London Number Theory SeminarPrevious Seminars

#### The London Number Theory Seminar is hosted in rotation for one semester each by UCL, IC and KCL.

The programme for

In Summer 2017, the seminar was hosted by Kings College and was organised by Nadav Yesha, Julien Hauseux and Wansu Kim.

26/04/17 Rebecca Bellovin (Imperial)
Title: Local $\varepsilon$-isomorphisms in families
Abstract: Given a representation of $Gal_{Q_p}$ with coefficients in a $p$-adically complete local ring $R$, Fukaya and Kato have conjectured the existence of a canonical trivialization of the determinant of a certain cohomology complex. When $R=Z_p$ and the representation is a lattice in a de Rham representation, this trivialization should be related to the $\varepsilon$-factor of the corresponding Weil--Deligne representation. Such a trivialization has been constructed for certain crystalline Galois representations, by the work of a number of authors. I will explain how to extend these trivializations to certain families of crystalline Galois representations. This is joint work with Otmar Venjakob.

Title: Arithmetic of hyperelliptic curves over local fields
Abstract: Let $C:y^2 = f(x)$ be a hyperelliptic curve over a local field $K$ of odd residue characteristic. I will explain how several arithmetic invariants of the curve and its Jacobian, including its potential stable reduction, Galois representation and (in the semistable case) Tamagawa numbers, can be simply extracted from combinatorial data coming from the roots of $f(x)$. This is joint work with Tim Dokchitser, Celine Maistret and Adam Morgan.

10/05/17 Chris Hughes (Univ of York)
Title: A new upper bound on Skewes' number
Abstract: The Prime Number Theorem tells us that the logarithmic integral, $li(x)$, is a good approximation to $\pi(x)$, the number of primes up to x. Numerically it always seems to be an overestimate, so $\pi(x)-li(x)$ is negative. The first point where this ceases to be the case is known as Skewes' number whose true value is as yet unknown. I will report on joint work with Chris Smith and Dave Platt, where we improve the best upper bound on Skewes' number.

Title: Pair correlation for quadratic polynomials mod 1.
Abstract: It is an open question whether the fractional parts of nonlinear polynomials at integers have the same fine-scale statistics as a Poisson point process. We provide explicit Diophantine conditions on the coefficients of degree 2 polynomials under which the limit of an averaged pair correlation density is consistent with the Poisson distribution, using a recent effective Ratner equidistribution result on the space of affine lattices due to Strömbergsson. This is joint work with Jens Marklof.

24/05/17 Stefano Vigni (Università di Genova)
Title: A Gross-Zagier formula for a certain anticyclotomic p-adic L-function of a rational elliptic curve.
Abstract: Let E be a (semistable) rational elliptic curve of conductor N, let K be an imaginary quadratic field satisfying a "Heegner hypothesis" relative to N and let p be a prime of split multiplicative reduction for E that splits in K. Following a recipe proposed by Bertolini and Darmon, I will define a p-adic L-function L_p(E/K) in terms of distributions of Heegner points on Shimura curves that are rational over the anticyclotomic Z_p-extension of K. The "special value" of L_p(E/K) is 0, and I will sketch a proof of a Gross-Zagier formula for the first derivative of L_p(E/K) involving a Heegner point over K and a p-adic L-invariant of E à la Mazur-Tate-Teitelbaum. The strategy is based on level raising arguments and Jochnowitz-type congruences. This is joint work (in progress) with Rodolfo Venerucci.

31/5/17 Wushi Goldring (Stockholm Uni)
Title: Geometry engendered by G-Zips: Shimura varieties and beyond.
Abstract: Moonen, Pink, Wedhorn and Ziegler initiated a theory of G-Zips, which is modeled on the de Rham cohomology of varieties in characteristic p>0 "with G-structure", where G is a connected reductive F_p-group. Building on their work, when X is a good reduction special fiber of a Hodge-type Shimura variety, it has been shown that there exists a smooth, surjective morphism \zeta from X to a quotient stack G-Zip^{\mu}. When X is of PEL type, the fibers of this morphism recover the Ekedahl-Oort stratification defined earlier in terms of flags by Moonen. It is commonly believed that much of the geometry of X lies beyond the structure of \zeta.

I will report on a project, initiated jointly with J.-S. Koskivirta and developed further in joint work with Koskivirta, B. Stroh and Y. Brunebarbe, which contests this common view in two stages: The first consists in showing that fundamental geometric properties of X are explained purely by means of \zeta (and its generalizations). The second is that, while these geometric properties may appear to be special to Shimura varieties, the G-Zip viewpoint shows that they hold much more generally, for geometry engendered by G-Zips: Any scheme Z equipped with a morphism to GZip^{\mu} satisfying some general scheme-theoretic properties. To illustrate our program concretely, I will describe results and conjectures regarding two basic geometric questions about X, Z: (i) Which automorphic vector bundles on X, Z admit global sections? (ii) Which of these bundles are ample? As a corollary, we also deduce old and new results over the complex numbers. Question (i) was inspired by a conjecture of F. Diamond on Hilbert modular forms mod p.

7/6/17 Jens Marklof (Univ of Bristol)
Title: Higher dimensional Steinhaus problems.
Abstract: The three gap theorem (or Steinhaus conjecture) asserts that there are at most three distinct gap lengths in the fractional parts of the sequence $\alpha, 2\alpha,\ldots, N\alpha$, for any integer $N$ and real number $\alpha$. This statement was proved in the 1950s independently by various authors. In this talk I will explain a different approach, which is based on the geometry of the space of two-dimensional Euclidean lattices (with Andreas Strombergsson, Amer. Math. Monthly, in press). This approach can in fact be generalised to deal with analogous higher dimensional Steinhaus problems for gaps in the fractional parts of linear forms. Here we are able to shed new light on a question of Erdos, Geelen and Simpson, proving the existence of parameters for which the number of distinct gaps is unbounded (joint work with Alan Haynes).

14/6/17 Beth Romano (Univ of Cambridge)
Title: On the arithmetic of simple singularities of type E.
Abstract: Given a simply laced Dynkin diagram, one can use Vinberg theory of graded Lie algebras to construct a family algebraic curves. In the case when the diagram is of type E7 or E8, Jack Thorne and I have used the relationship between these families of curves and their associated Vinberg representations to gain information about integral points on the curves. In my talk, I'll focus on the role Lie theory plays in the construction of the curves and in our proofs.

21/6/17 Samit Dasgupta (UC Santa Cruz)
Title: On the characteristic polynomial of Gross's regulator matrix.
Abstract: Let $F$ be a totally real field and $\chi$ a totally odd character of $F$. Gross conjectured that the leading term of the Deligne-Ribet $p$-adic $L$-function associated to $\chi$ at $s=0$ is equal to a $p$-adic regulator of $p$-units in the extension of $F$ cut out by $\chi$. I recently proved this result in joint work with Mahesh Kakde and Kevin Ventullo. The topic of this talk is a refinement of Gross's conjecture. I will propose an analytic formula for the characteristic polynomial of Gross's regulator matrix, rather than just its determinant. The formula is given in terms of the Eisenstein cocycle and in fact applies (conjecturally) to give all the principal minors of Gross's matrix. For the diagonal entries, the conjecture overlaps with the conjectural formula presented in prior work. This is joint work with Michael Spiess.

28/6/17 Morten Risager (Uni of Copenhagen)
Title: Arithmetic statistics of modular symbols.
Abstract: Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve L-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols. We prove a refined version of this conjecture.
This is joint work with Yiannis Petridis.

In Spring 2017, the seminar was hosted by UCL and was organised by Sarah Zerbes.

11/01/17 Chris Skinner (Princeton)
Title: Recent progress on the Iwasawa theory of elliptic curves and modular forms.
Abstract: This talk will describe some of the recent work on the Iwasawa theory of modular forms (at both ordinary and non-ordinary primes) with an emphasize on the strategy of proof, which involves two different main conjectures.

18/01/17 Jack Lamplugh (UCL)
Title: An Euler system for a pair of CM modular forms.
Abstract: Given a pair of modular forms and a prime p, Lei-Loeffler-Zerbes have constructed an Euler system for the tensor product of the p-adic Galois representations attached to each of the forms. When the forms have CM by distinct imaginary quadratic fields, this representation is induced from a character $\chi$ over an imaginary biquadratic field F. I will explain how one can use this Euler system to obtain upper bounds for Selmer groups associated to $\chi$ over the $\mathbf{Z}_p^3$-extension of F.

Title: The Hasse norm principle for abelian extensions
Abstract: Let $L/K$ be an extension of number fields and let $J_L$ and $J_K$ be the associated groups of ideles. Using the diagonal embedding, we view $L^\times$ and $K^\times$ as subgroups of $J_L$ and $J_K$ respectively. The norm map $N: J_L\to J_K$ restricts to the usual field norm $N: L^\times\to K^\times$ on $L^\times$. Thus, if an element of $K^\times$ is a norm from $L^\times$, then it is a norm from $J_L$. We say that the Hasse norm principle holds for $L/K$ if the converse holds, i.e. if every element of $K^\times$ which is a norm from $J_L$ is in fact a norm from $L^\times$. The original Hasse norm theorem states that the Hasse norm principle holds for cyclic extensions. Biquadratic extensions give the smallest examples for which the Hasse norm principle can fail. One might ask, what proportion of biquadratic extensions of $K$ fail the Hasse norm principle? More generally, for an abelian group $G$, what proportion of extensions of $K$ with Galois group $G$ fail the Hasse norm principle? I will describe the finite abelian groups for which this proportion is positive. This involves counting abelian extensions of bounded discriminant with infinitely many local conditions imposed, which is achieved using tools from harmonic analysis. This is joint work with Christopher Frei and Daniel Loughran.

1/2/17 Atsuhira Nagano (KCL/Waseda University)
Title: K3 surfaces and a construction of a Shimura variety
Abstract: In old times, elliptic modular functions appeared in the study of elliptic curves. They are applied to the construction of class fields. (This is classically called Kronecker's Jugendtraum.) K3 surfaces are 2-dimensional analogy of elliptic curves. In this talk, the speaker will present an extension of the classical result by using K3 surfaces. Namely, we will obtain Hilbert modular functions via the periods of K3 surfaces and construct a certain model of a Shimura variety explicitly.

8/2/17 Christian Johansson (University of Cambridge)
Title: Integral models for eigenvarieties
Abstract: I will discuss a construction of integral models of eigenvarieties using a generalization of the overconvergent distribution modules of Ash and Stevens, and their relation to recent work of Andreatta-Iovita-Pilloni and Liu-Wan-Xiao on the geometry of the Coleman-Mazur eigencurve near the boundary of weight space. This is joint work with James Newton.

15/2/17 (note room change: Archaeology, Room G6)
Alan Lauder (Oxford)
Title: Stark points on elliptic curves and modular forms of weight one
Abstract: I shall discuss some work with Henri Darmon and Victor Rotger on the explicit construction of points on elliptic curves. The elliptic curves are defined over $\mathbb{Q}$, and the points over fields cut out by Artin representations attached to modular forms of weight one.

22/2/17 (note room change: Archaeology, Room G6)
Erick Knight (Harvard/Bonn)
Abstract: In this talk, I will construct a p-adic Jacquet-Langlands correspondence, which is a correspondence between Banach space representations of GL2(Qp) and Banach space representations of the unit group of the quaternion algebra D over Qp. The correspondence satisfies local-global compatibility with the completed cohomology of Shimura curves, as well as a compatibility with the classical Jacquet-Langlands correspondence, in the sense that the $D^\times$ representations can often be shown to have the expected locally algebraic vectors.

1/3/17 Giuseppe Ancona (Université de Strasbourg)
Title: Standard conjectures for abelian fourfolds
Abstract: Let X be a smooth projective variety and V be the finite dimensional Q vector space of algebraic cycles on X modulo numerical equivalence. Grothendieck defined a quadratic form on V (basically using the intersection product) and conjectured that it is positive definite. This conjecture is a formal consequence of Hodge Theory in characteristic zero, but almost nothing is known in positive characteristic. Instead of studying this quadratic form at the non archimedean place (the signature) we will study it at the p-adic places. It turns out that this question is more treatable. Moreover, using a product formula formula, the p-adic information will give us non trivial informations on the non archimedean place. For instance we will show the original conjecture when X is an abelian variety of dimension 4.

8/3/17 Céline Maistret (Warwick)
Title: Parity of ranks of abelian surfaces
Abstract: Let K be a number field and A/K an abelian surface (dimension 2 analogue of an elliptic curve). By the Mordell-Weil theorem, the group of K-rational points on A is finitely generated and as for elliptic curves, its rank is predicted by the Birch and Swinnerton-Dyer conjecture. A basic consequence of this conjecture is the parity conjecture: the sign of the functional equation of the L-series determines the parity of the rank of A/K. Under suitable local constraints and finiteness of the Shafarevich-Tate group, we prove the parity conjecture for principally polarized abelian surfaces. We also prove analogous unconditional results for Selmer groups.

15/3/17 Aurel Page (Warwick)
Title: Computing the homotopy type of compact arithmetic manifolds
Abstract: Cohomology of arithmetic manifolds equipped with the action of Hecke operators provides concrete realisations of automorphic representations. I will present joint work with Michael Lipnowski where we describe and analyse an algorithm to compute such objects in the compact case. I will give a gentle introduction to the known case of dimension 0, sketch ideas and limitations of previous algorithms in small dimensions, and then explain some details and ideas from the new algorithm.

22/3/17 Otto Overkamp (Imperial)
Title: Finite descent obstruction and non-abelian reciprocity
Abstract: For a nice algebraic variety X over a number field F, one of the central problems of Diophantine Geometry is to locate precisely the set X(F) inside X(A_F), where A_F denotes the ring of adeles of F. One approach to this problem is provided by the finite descent obstruction, which is defined to be the set of adelic points which can be lifted to twists of torsors for finite étale group schemes over F on X. More recently, Kim proposed an iterative construction of another subset of X(A_F) which contains the set of rational points. In this paper, we compare the two constructions. Our main result shows that the two approaches are equivalent.

The seminar will be preceded by the Study Groups, and this term there are two of them, one on p-adic integration running 1300-1415 and one on p-adic Hodge theory running 1430-1600.

In Autumn 2016, the seminar was hosted by Imperial and was organised by Stephane Bijakowski, Olivier Taibi and Rebecca Bellovin.

5/10/16 James Newton (Kings)
Title: Patching and the completed homology of locally symmetric spaces.
Abstract: I will explain a variant of Taylor--Wiles patching which applies to the completed homology of locally symmetric spaces for $\mathrm{PGL}(n)$ over a CM field. Assuming some natural conjectures about completed homology, I will describe some applications of our construction to the study of Galois representations and (p-adic) automorphic forms. This is joint work with Toby Gee.

12/10/16 Jean-Stefan Koskivirta (Imperial)
Title: Generalized Hasse invariants and some applications
Abstract: This talk is a report on a paper with Wushi Goldring. If A is an abelian variety over a scheme S of characteristic p, the isomorphism class of the p-torsion gives rise to a stratification on S. When it is nonempty, the ordinary stratum is open and the classical Hasse invariant is a section of the p-1 power of the Hodge bundle which vanishes exactly on its complement. In this talk, we will explain a group-theoretical construction of generalized Hasse invariants based on the stack of G-zips introduced by Pink, Wedhorn, Ziegler Moonen. When S is the good reduction special fiber of a Shimura variety of Hodge-type, we show that the Ekedahl-Oort stratification is principally pure. We apply Hasse invariants to attach Galois representations to certain automorphic representations whose archimedean part is a limit of discrete series, and to study systems of Hecke-eigenvalues that appear in coherent cohomology.

19/10/16 Andrea Bandini (Università degli Studi di Parma)
Title: Stickelberger series and Iwasawa Main Conjecture for $\mathbb{Z}_p^\infty$-extensions of function fields
Abstract: Let $F:=\mathbb{F}_q(\theta)$ and let $\mathfrak{p}$ be a prime of $A:=\mathbb{F}_q[\theta]$ ($q=p^r$ and $p$ a prime). Let $\mathcal{F}_{\mathfrak{p}}/F$ be the $\mathfrak{p}$-cyclotomic $\mathbb{Z}_p^\infty$-extension of $F$ generated by the $\mathfrak{p}^\infty$-torsion of the Carlitz module and let $\Lambda$ be the associated Iwasawa algebra. We give an overview of the Iwasawa theory for the $\Lambda$-module of divisor class groups and then define a Stickelberger series in $\Lambda[[u]]$, whose specializations enable us to prove an Iwasawa Main Conjecture for this setting. As an application we obtain a close analogue of the Ferrero-Washington theorem for $\mathcal{F}_{\mathfrak{p}}$. (Joint work with Bruno Anglès, Francesc Bars and Ignazio Longhi)

Title: Sansuc's formula and Tate global duality (d'apr\es Rosengarten).
Abstract: Tamagawa numbers are canonical (finite) volumes attached to smooth connected affine groups $G$ over global fields $k$; they arise in mass formulas and local-global formulas for adelic integrals. A conjecture of Weil (proved long ago for number fields, and recently by Lurie and Gaitsgory for function fields) asserts that the Tamagawa number of a simply connected semisimple group is equal to 1; for special orthogonal groups this expresses the Siegel Mass Formula. Sansuc pushed this further (using a lot of class field theory) to give a formula for the Tamagawa number of any connected reductive $G$ in terms of two finite arithmetic invariants: its Picard group and degree-1 Tate-Shafarevich group.

Over number fields it is elementary to remove the reductivity hypothesis from Sansuc's formula, but over function fields that is a much harder problem; e.g., the Picard group can be infinite. Work in progress by my PhD student Zev Rosengarten is likely to completely solve this problem. He has formulated an alternative version, proved it is always finite, and established the formula in many new cases. We will discuss some aspects of this result, including one of its key ingredients: a generalization of Tate local and global duality to the case of coefficients in any positive-dimensional (possibly non-smooth) affine algebraic $k$-group scheme and its (typically non-representable) ${\rm{GL}}_1$-dual sheaf for the fppf topology.

2/11/16 Joe Kramer-Miller (UCL)
Title: F-isocrystals with infinite monodromy
Abstract: Let $U$ be a smooth geometrically connected affine curve over $\mathbb{F}_p$ with compactification $X$. Following Dwork and Katz, a $p$-adic representation $\rho$ of $\pi_1(U)$ corresponds to an etale F-isocrystal. By work of Tsuzuki and Crew an F-isocrystal is overconvergent precisely when $\rho$ has finite monodromy. However, in practice most F-isocrystals arising geometrically are not overconvergent and instead have logarithmic decay at singularities (e.g. characters of the Igusa tower over a modular curve). We give a Galois-theoretic interpretation of these log decay F-isocrystals in terms of asymptotic properties of higher ramification groups.

9/11/16 Carl Wang Erickson (Imperial)
Title: Pseudorepresentations and the Eisenstein ideal
Abstract: In his landmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. He proved a great deal about these congruences, and also posed a number of questions: how big is the space of cusp forms that are congruent to the Eisenstein series? How big is the extension generated by their coefficients? In joint work with Preston Wake, we give an answer to these questions using the deformation theory of Galois pseudorepresentations. The answer is intimately related to the algebraic number theoretic interactions between the primes N and p, and is given in terms of cup products (and Massey products) in Galois cohomology.

16/11/16 Dimitar Jetchev (EPFL)
Title: The p-part of the Birch and Swinnerton-Dyer Conjecture for Elliptic Curves of Analytic Rank One
Abstract: I will explain the recent proof of the p-part of the Birch and Swinnerton-Dyer Conjectural formula for elliptic curves over Q of analytic rank one. The proof is based on choosing a suitable parametrization of the elliptic curve with a Shimura curve, using Kolyvagin's Euler system method to get the upper bounds and an anticyclotomic Iwasawa main conjecture as well as a control theorem to get the lower bounds. This is joint work with Chris Skinner and Xin Wan.

23/11/16 Macarena Peche Irissarry (ENS Lyon)
Title: Reduction of $G$-ordinary crystalline representations with $G$-structure
Abstract: Fontaine's $D_{\mathrm{cris}}$ functor allows us to associate an isocrystal to any crystalline representation. For a reductive group $G$, we study the reduction of lattices inside a germ of crystalline representations with $G$-structure $V$ to lattices (which are crystals) with $G$-structure inside $D_{\mathrm{cris}}(V)$. Using Kisin modules theory, we give a description of this reduction in terms of $G$, in the case where the representation $V$ is ($G$-)ordinary. In order to do that, first we need to generalize Fargues' construction of the Harder-Narasimhan filtration for $p$-divisible groups to Kisin modules.

30/11/16 Valentin Hernandez (Paris VI)
Title: $\mu$-ordinary Hasse invariants and the canonical filtration of a p-divisible group.
Abstract: In his 1973 paper, Katz constructed overconvergent modular forms on the modular curve geometrically, using the Hasse invariant and Lubin’s Theorem on the canonical subgroup of an elliptic curve. Many improvements have since been made on these constructions on many Shimura varieties, but this approach is now well known only when the ordinary locus is non-empty. I will try to explain how to get rid of this assumption, and detail the construction of a replacement for the Hasse invariant and the construction of the canonical filtration focusing on the local analogue of the Picard modular surface.

7/12/16 Note that this week the seminar is in 130.
Joaquin Rodrigues (UCL)
Abstract: Let p be a prime number. We will discuss how to associate, to a modular form f of level N, a (partial) p-adic L-function interpolating special values of the complex L-function of f. This construction is based upon Kato's Euler system and the theory of $(\varphi, \Gamma)$-modules. We will also discuss a functional equation on the Iwasawa theory for Galois representations of dimension 2 and how this gives, on the one hand a functional equation for our p-adic L-function, and on the other hand results on Kato's local epsilon conjecture 2-dimensional representations.

14/12/16 Jaclyn Lang (Paris XIII)
Title: Images of Galois representations associated to Hida families
Abstract: We explain a sense in which Galois representations associated to non-CM Hida families have large images. This is analogous to results of Ribet and Momose for Galois representations associated to classical modular forms. In particular, we show how extra twists of the Hida family decreases the size of the image.

In Summer 2016, the seminar was hosted by KCL and was organised by Mahesh Kakde.

27/4/16 Fernando Shao (Oxford)
Title: Vinogradov's three primes theorem with almost twin primes
Abstract: The general theme of this talk is about solving linear equations in sets of number theoretic interest. Specifically I will discuss the problem with the linear equations being N = x+y+z (for a fixed large N) and the set being "almost twin primes". The focus will be on the underlying ideas coming from both additive combinatorics and sieve theory. This is joint work with Kaisa Matomaki.

4/5/16 No seminar because of conference RandomWavesInLondon.

11/5/16 Oleksiy Klurman (Universite de Montreal/UCL)
Title: Correlations of multiplicative functions and applications.
Abstract: A deep problem in analytic number theory is to understand correlations of general multiplicative functions. In this talk, we derive correlations formulas for so-called bounded "pretentious" multiplicative functions. This has a number or desirable consequences. First, we characterize all multiplicative functions $f:\mathbb{N}\to \{-1,1\}$ with bounded partial sums. This answers a question of Erd\H{o}s from 1957 in the form conjectured by Tao. Second, we show that if the average of the first divided difference of multiplicative function is zero, then either $f(n)=n^s$ for $\operatorname{Re}(s)<1$ or $|f(n)|$ is small on average. This settles an old conjecture of K\'atai. If time permits, we discuss some further applications to the related problems.

18/5/16 Giovanni Rosso (Cambridge)
Title: Trivial zero for a p-adic L-function associated with Siegel forms
Abstract: We shall begin with an introduction to L-functions in arithmetic, their p-adic interpolation and trivial zero. We shall state a conjecture of Greenberg and Benois which predicts the order and the leading coefficient of p-adic L-functions when trivial zeros appear. We shall then explain how one calculates the first derivative of the standard p-adic L-function of an ordinary Siegel form with level at p.

25/5/16 Gergely Zábrádi (Eötvös Loránd)
Title: Smooth mod p^n representations and direct powers of Galois groups.
Abstract: Let G be a Qp-split reductive group with connected centre and Borel subgroup B=TN. We construct a right exact functor D from the category of smooth modulo p^n representations of B to the category of projective limits of continuous mod p^n representations of a direct power of the absolute Galois group Gal(Qpbar/Qp) of Qp indexed by the set of simple roots. The objects connecting the two sides are (phi,Gamma)-modules over a multivariable (commutative) Laurent series ring which correspond to the Galois side via an equivalence of categories. Parabolic induction from a subgroup P = L_P N_P amounts to the extension of the representation on the Galois side to the copies of Gal(Qpbar/Qp) indexed by the simple roots alpha not contained in the Levi component L_P using the action of the image of the cocharacter dual to alpha and local class field theory. D is exact and yields finite dimensional representations on the category SP of finite length representations with subquotients of principal series as Jordan-Hölder factors. Using the G-equivariant sheaf of Schneider, Vigneras, and the author on the flag variety G/B corresponding to the Galois representation we show that D is fully faithful on the full subcategory of SP with Jordan-Hölder factors isomorphic to irreducible principal series. Breuil has (preliminary) conjectures for the values of D at certain representations of GL_n(Qp) built out from some mod p Hecke isotypic subspaces of global automorphic representations.

1/6/16 Trevor Wooley (Bristol)
Title: Sub-convexity in certain Diophantine problems via the circle method.
Abstract: The sub-convexity barrier traditionally prevents one from applying the Hardy-Littlewood (circle) method to Diophantine problems in which the number of variables is smaller than twice the inherent total degree. Thus, for a homogeneous polynomial in a number of variables bounded above by twice its degree, useful estimates for the associated exponential sum can be expected to be no better than the square-root of the associated reservoir of variables. In consequence, the error term in any application of the circle method to such a problem cannot be expected to be smaller than the anticipated main term, and one fails to deliver an asymptotic formula. There are perishingly few examples in which this sub-convexity barrier has been circumvented, and even fewer having associated degree exceeding two. In this talk we review old and more recent progress, and exhibit a new class of examples of Diophantine problems associated with, though definitely not, of translation-invariant type.

6th and 7th June -- London-Paris Number Theory Seminar.

Title: Gaussian and non-Gaussian behaviour of character sums
Abstract: Davenport and Erdos, and more recently Lamzouri, have investigated the distribution of short character sums $\sum_{x < n \leq x+H} \chi(n)$ as $x$ varies, for a fixed non-principal character $\chi$ modulo $q$. In particular, Lamzouri conjectured that these sums should have a Gaussian limit distribution (real or complex according as $\chi$ is real or complex) provided $H=H(q)$ satisfies $H \rightarrow\infty$ but $H = o(q/\log q)$. I will describe some work in progress in connection with this conjecture. In particular, I will try to explain that the conjecture cannot quite be correct (one need not have Gaussian behaviour for $H$ as large as $q/\log q$), but on the other hand one should see Gaussian behaviour for even larger $H$ for most characters.

15/6/16 Min Lee (Bristol)
Title: Effective equidistribution of primitive rational points on expanding horospheres
Abstract: The limit distribution of primitive rational points on expanding horospheres on SL(n,Z)\SL(n, R) has been derived in the recent work by M. Einsiedler, S. Mozes, N. Shah and U. Shapira. For n=3, in our joint project with Jens Marklof, we prove the effective equidistribution of q-primitive points on expanding horospheres as q tends to infinity.

22/6/16 Maria Valentino (King's)
Title: On the diagonalizability of the Atkin U-operator for Drinfeld cusp forms
Abstract: In this talk we shall begin with an introduction to Drinfeld cusp forms for arithmetic subgroups using Teitelbaum's interpretation as harmonic cocycles. We shall then address the problem of the diagonalizability of the function field analogous of the Atkin U-operator carrying the Hecke action over harmonic cocycles.

29/6/16 Nina Snaith (Bristol)
Title: Unearthing random matrix theory in the statistics L-functions: the story of Beauty and the Beast
Abstract: There has been very convincing numerical evidence since the 1970s that the positions of zeros of the Riemann zeta function and other L-functions show the same statistical distribution (in the appropriate limit) as eigenvalues of random matrices. Proving this connection, even in restricted cases, is difficult, but if one accepts the connection then random matrix theory can provide unique insight into long-standing questions in number theory. I will give some history of the attempt to prove the connection, as well as propose that the way forward may be to forgo the enticing beauty of the determinantal formulae available in random matrix theory in favour of something a little less elegant (work with Brian Conrey and Amy Mason)

In Spring 2016, the seminar was hosted by UCL and was organised by Sarah Zerbes and Andrew Granville.

January 13
Tobias Berger (Sheffield)
Title: Oddness of Galois representations and criticality of L-values

Abstract: Following Ribet's seminal 1976 paper there have been many results employing congruences between stable cuspforms and lifted forms to construct non-split extensions of Galois representations. This strategy can be extended to construct elements in the Bloch-Kato Selmer groups of general Â±-Asai representations. I will explain how suitable congruences between automorphic forms over a CM field, whose associated Galois representations are totally odd polarizable, always give rise to elements in the Selmer group for exactly the Asai representation (+ or -) that is critical in the sense of Deligne. In addition, I will discuss consequences for Fontaine-Mazur style conjectures.

January 20
Eyal Goren (McGill)
Title: Picard modular surfaces in positive characteristic

Abstract: The theory of modular curves, their integral models and modular forms on them is well-developed, and had been used in many spectacular applications. I will recall some of the features that are relevant to my talk. Motivated by the state of affairs for curves, we have been studying unitary Shimura varieties in positive characteristic and, in particular, the Picard modular surfaces that are associated to a unitary group of signature (2,1). These are moduli spaces for abelian threefolds equipped with an action of an imaginary quadratic field. I will explain what we currently know about their geometry modulo a prime p (building on work by Bellaiche, Vollaard, Bultel-Wedhorn and borrowing ideas from G. Boxer and the theory of modular curves). To the extent time allows, I will discuss Hecke correspondences at p and the rather complicated picture we face. This is joint work with E. De Shalit (Hebrew University).

January 27
There will be two seminars, from 2:45-4:15pm and from 4:45-5:45.
Sofia Lindqvist (Oxford)
Title: Even moments of random multiplicative functions

Abstract: We give an asymptotic formula for the even moments of a sum of multiplicative Steinhaus or Rademacher random variables. This is obtained by expressing the sum as a multiple contour integral from which the asymptotic behaviour can be extracted. The result was obtained independently by Harper, Nikeghbali, and Radziwill by using a result of La Breteche. We also give an asymptotic relationship between the Steinhaus even moments and the even moments of a truncated characteristic polynomial of a unitary matrix, which extends earlier work of Conrey and Gamburd. The talk is based on joint work with Winston Heap.

Guhan Harikumar (UCL)
Title: Darmon cycles and the Kohnen-Shintani lifting

Abstract: We will first recall the theory of Darmon cycles due to V. Rotger and M. Seveso. These are a higher weight analogue of Stark-Heegner points. Then, we will show how these Darmon cycles are related to (a p-adic family of) half-integral weight modular forms. The relation follows by the p-adic interpolation of a well known formula of Waldspurger.

February 3
Yukako Kezuka (Cambridge)
Title: On the main conjecture of Iwasawa theory for certain elliptic curves with complex multiplication

Abstract: Let E be a quadratic twist of the elliptic curve X_0(49), so that E has complex multiplication by the ring of integers of Q(sqrt(-7)). Using Iwasawa theory, Gonzalez-Aviles and Rubin proved that if L(E/Q,1) is nonzero, then the full Birch--Swinnerton-Dyer conjecture holds for E. We will consider a more general case: Take p to be any prime which is congruent to 7 modulo 8, and set K= Q(sqrt(-p)). We will discuss the main conjecture of Iwasawa theory for an infinite family of elliptic curves which are defined over the Hilbert class field of K with complex multiplication by the ring of integers of K.

February 10
THIS SEMINAR WILL BE FROM 2:45-3:45PM (INSTEAD OF THE SECOND STUDY GROUP), AS THERE IS ANDREW GRANVILLE'S INAUGURAL LECTURE FROM 4-5PM.

Eugen Hellmann (Bonn)
Title: On companion points on eigenvareties

Abstract: In the theory of p-adic modular forms (or more generally p-adic automorphic forms) the phenomenon occurs that there are non-classical and classical forms that have the same system of Hecke-eigenvalues. This phenomenon has an explanation in terms of the associated Galois representations. Namely certain p-adic Hodge-structures (so called triangulations) degenerate in the corresponding families of Galois representations. We prove various results about the space of corresponding Galois representations and relate them to questions about companion points.

February 17
Dan Loughran (Hannover)
Title: Fibrations with few rational points

Abstract: In this talk, we study the problem of counting the number of varieties in fibrations over projective spaces which contain a rational point. We obtain geometric conditions that force very few of the varieties in the family to contain a rational point, in a precise quantitative sense. This generalises the special case of conic bundles treated by Serre. This is joint work with Arne Smeets.

February 24
Min Lee (Bristol)
Title: The Selberg trace formula as a Dirichlet series

Abstract: We explore the idea of Conrey and Li of presenting the Selberg trace formula for Hecke operators, as a Dirichlet series. We enhance their work in few ways and present several applications of our formula. This is a joint work with Andrew Booker.

March 2
Roger Heath-Brown (Oxford)
Title: Vinogradov's Mean Value and the Riemann Zeta-Function

Abstract: We shall describe the recent results of Wooley, and of Bourgain Demeter and Guth, on Vinogradov's mean value, and how they lead to substantial improvements in estimates for the Riemann Zeta-function.

March 9
There will be two seminars, from 2:45-3:45pm and from 4:45-5:45.
Sary Drappeau (Marseille)
Title: Divisors, additive functions and smooth numbers

Abstract: The distribution of divisors of an integer n can be studied through the distribution of the random variable D_n := (log d)/log n where d is chosen uniformly at random from divisors of n. Out of many interesting aspects of the sequence (D_n), one may ask about its possible convergence in law, as n tends to infinity along various sequences of integers. For instance, a classical result of Deshouillers, Dress and Tenenbaum shows the convergence of the Cesaro mean of the distribution functions of D_n. It appears however that for generic n, the situation is very erratic. In this talk, we will consider the case where n is assumed not to have any large prime factor (for a certain meaning of "large"). In this situation, a central limit theorem is expected to hold for D_n, and is established on average. This will lead us to various topics in probabilistic number theory.

Tim Browning (Bristol)
Title: Rational points on varieties via counting

Abstract: I will discuss some recent progress on the Brauer-Manin obstruction for rational points on algebraic varieties, before showing how counting arguments from analytic number theory can be used to study strong approximation for a special family of varieties defined by norm forms. This allows the resolution of a conjecture of Colliot-Thelene about the sufficiency of the Brauer-Manin obstruction for varieties admitting a suitable fibration. This is joint work with Damaris Schindler.

March 16
Miguel Walsh (Oxford)
Title: On the polynomial method in number theory

Abstract: n broad terms, the polynomial method is the idea of understanding a set of points, for example in euclidean space, by studying the properties of polynomials vanishing on that set. We'll discuss some applications of this method in problems related to number theory and combinatorics, including connections with sieve theory, the study of rational points on curves and the incidence geometry of algebraic varieties.

23/3/16 The seminar this week will be at Kings College, at 2pm, in S4.23
Samit Dasgupta (UCSC)
Title: On the Gross-Stark conjecture
Abstract: In 1980, Gross conjectured a formula for the expected leading term at $s=0$ of the Deligne--Ribet $p$-adic $L$-function associated to a totally even character $\psi$ of a totally real field $F$. The conjecture states that this value is equal to a $p$-adic regulator of units in the abelian extension of $F$ cut out by $\psi \omega^{-1}$. In this talk I will describe a proof of Gross's conjecture. Our methods build on our previous joint work with Darmon and Pollack and work of Ventullo, which together prove this conjecture in the rank 1 case. The current work is joint with Mahesh Kakde and Kevin Ventullo.

29/3/16 Seminar is on a Tuesday 3pm in S4.36 at Kings
Anish Ghosh (Tata Institute)
Title: Values of quadratic forms at integer points
Abstract: I will discuss Margulis' resolution of a long open conjecture of Oppenheim on values of irrational indefinite quadratic forms and some recent quantitative analogues. The latter is joint work with Alexander Gorodnik and Amos Nevo.

In Autumn 2015, the seminar was hosted by Imperial College and was organised by David Helm.

07/10/15 Martin Orr (Imperial)
Title: A bound for rational representations of isogenies in the fundamental set
Abstract: Let z and z' be two points in the standard fundamental set in the upper half-plane. If the corresponding elliptic curves are related by an isogeny of degree N, then there is a 2x2 matrix with integer coefficients and determinant N which maps z to z'. As an ingredient in their work on unlikely intersections, Habegger and Pila proved that the entries of this matrix are bounded by a uniform polynomial in N. I will discuss the generalisation of this result to moduli of abelian varieties and beyond, to Riemannian symmetric spaces of non-compact type.

14/10/15 Stephane Bijakowski (Imperial)
Title: The partial degrees of the canonical subgroup
Abstract: If the Hasse invariant of a p-divisible group is small enough, then one can construct a canonical subgroup inside its p-torsion. I will first present an alternative approach to this problem, assuming the existence of a subgroup satisfying some simple conditions. A key property is the relation between the Hasse invariant and the degree of the canonical subgroup. When one considers a p-divisible group with extra structures, more information is available. I will define the partial Hasse invariants, the partial degrees, and relate them for the canonical subgroup.

21/10/15 Yiwen Ding (Imperial)
Title: L-invariants and local-global compatibility for GL2
Abstract: Let F be a totally real number field, w a prime of F above p, V a 2-dimensional p-adic representation of the absolute Galois group G_F of F which appears in etale cohomology of quaternion Shimura curves. When the restriction Vw of V to the decomposition group of G_F at w is semi-stable non-crystalline in Fontaine's sense, we can associate to Vw the so-called Fontaine-Mazur L-invariants, which are invisible in classical local Langlands correspondance. We show that these L-invariants can be found in the completed cohomology group of Shimura curves.

28/10/15 Rob Kurinczuk (Bristol)
Title: Cuspidal l-modular representations of classical p-adic groups
Abstract: For a classical groups (unitary, special orthogonal, symplectic) over locally compact non-archimedean fields of odd residual characteristic p, Shaun Stevens has developed an approach to studying their (smooth) complex representations based on the theory of types of Bushnell and Kutzko. I will describe some joint work with Shaun Stevens, in which we relate positive level cuspidal representations in Stevens' construction to level zero cuspidal representations in certain associated groups and consider a generalisation to modular representations in characteristic prime to p.

04/11/15 Paul Ziegler
Title: p-kernels occurring in isogeny classes of p-divisible groups
Abstract: I will give a criterion which allows to determine, in terms of the combinatorics of the root system A_n, which p-kernels occur in a given isogeny class of p-divisible groups over an algebraically closed field of positive characteristic. This question is related to the relationship between Newton and Ekedahl-Oort strata on reductions of Shimura varieties as well as the non-emptiness of affine Deligne-Lusztig varieties.

9/11/15 London-Paris Number Theory Seminar (in Paris)

11/11/15 Laurent Berger
Title: Iterated extensions and relative Lubin-Tate groups
Abstract: An important construction in p-adic Hodge theory is the 'field of norms' corresponding to an infinite extension K_infty/K. For the cyclotomic extension, it is possible to lift the field of norms to characteristic zero, and we can ask for which other extensions K_infty/K this is possible. The goal of this talk is to explain this question and discuss some partial answers. This involves p-adic dynamical systems, Coleman power series and relative Lubin-Tate groups.

18/11/15 Julien Hauseux (Kings)
Title: Parabolic induction and extensions
Abstract: Let G be a p-adic reductive group. We describe the extensions between admissible smooth mod p representations of G which are parabolically induced from supersingular representations of Levi subgroups. More precisely, we determine which extensions do not come from parabolic induction. In order to do so, we partially compute Emerton's delta-functor of derived ordinary parts on any parabolically induced representation of G. These computations work with mod p^n coefficients, thus some of the results on extensions can be lifted in characteristic 0 for admissible unitary continuous p-adic representations of G.

25/11/15 Arne Smeets (Imperial)
Title: Logarithmic good reduction and cohomological tameness
Abstract: I will discuss two notions of tameness for varieties defined over a field equipped with a discrete valuation, which are only interesting if the residual characteristic is positive: cohomological tameness, and logarithmic good reduction. The first notion is weaker than the second one (Nakayama). I will explain why these notions are equivalent in the case of abelian varieties; this can be seen as a logarithmic version of the Néron-Ogg-Shafarevich criterion (joint work with A. Bellardini). I will also discuss a cohomological trace formula for the tame monodromy operator, conjectured by Nicaise for cohomologically tame varieties, proven by the speaker for varieties with logarithmic good reduction.

2/12/15 Jacques Benatar (Kings)
Title: Goldbach versus de Polignac numbers
Abstract: We discuss the following statement, connecting two well-known conjectures. Either consecutive Goldbach numbers lie within a finite distance from one another or else the set of de Polignac numbers has full density in 2N.

9/12/15 Jennifer Balakrishnan (Oxford)
Title: Rational points and quadratic Chabauty
Abstract: Let C be a curve over the rationals of genus g at least 2. By Faltings' theorem, we know that C has finitely many rational points. When the Mordell-Weil rank of the Jacobian of C is less than g, the Chabauty-Coleman method can often be used to find these rational points through the construction of certain p-adic integrals. When the rank is equal to g, we can use the theory of p-adic height pairings to produce p-adic double integrals that allow us to find integral points on curves. In particular, I will discuss how to carry out this "quadratic Chabauty" method on hyperelliptic curves over number fields (joint work with Amnon Besser and Steffen Mueller) and present related ideas to find rational points on bielliptic genus 2 curves (joint work with Netan Dogra).

16/12/15 Daniel Skodlerack (Imperial)
Title: Cuspidal representations for inner forms of classical groups
Abstract: In this talk I present the strategy to classify the cuspidal irreducible representations of inner forms of classical groups, and I will give some remarks on the role of Endo-classes for classical groups.

In Summer 2015, the seminar was hosted by Kings College and was organised by Mahesh Kakde and Igor Wigman.

22/04/15 2.30 (in the usual place S4.23) Jack Shotton (Imperial)
Title: Local Galois deformation rings when l != p.
Abstract: Given a mod p representation of the absolute Galois group of Q_l, consider the universal framed deformation ring R parametrising its lifts. When l and p are distinct I will explain a relation between the mod p geometry of R and the mod p representation theory of GL_n(Z_l), that is parallel to the Breuil-Mézard conjecture in the l = p case. I will give examples and say something about the proof, which uses automorphy lifting techniques.

22/04/15 4.00 Christophe Breuil (Paris)
Title: Classicality on Eigenvarieties
Abstract: Let p be a prime number. It is expected that a p-adic overconvergent automorphic eigenform of classical weight (on a definite unitary group of arbitrary dimension) such that its associated Galois representation is crystalline at p should be classical. I will sketch a proof of many new cases of this conjecture. This is joint work with E. Hellmann and B. Schraen.

29/04/15 4.00 Victor Beresnevich (York)
Abstract: Real numbers badly approximable by rational numbers have been known for well over a century, thanks to continued fractions. In this talk I will discuss recent results in Diophantine approximation on manifolds that led to the proof of the existence of transcendental real numbers badly approximable by algebraic numbers of arbitrary fixed degree.

06/05/15 4.00 Wansu Kim
Title: Rapoport-Zink spaces of Hodge type and applications to Shimura varieties
Abstract: Rapoport-Zink spaces of (P)EL type are local analogues of Shimura varieties of PEL type. Examples include Lubin-Tate spaces and Drinfeld upper half spaces.
In this talk, we construct the "Hodge-type generalisation" of Rapoport-Zink spaces under the unramifiedness hypothesis, and apply it to the integral models of Hodge-type Shimura varieties. The new examples are Spin and orthogonal Rapoport-Zink spaces (of arbitrary rank) -- local analogues of Spin and orthogonal Shimura varieties.
We will start with the description of geometric points of "Hodge-type Rapoport-Zink spaces" and the completed local rings thereof. Some applications to the study of Hodge-type Shimura varieties will be given.

13/05/15 4.00 Anne-Maria Ernvall-Hytonen (Helsinki)
Title: On Baker type bounds and generalised transcendence measure (joint work with K. Leppälä and Tapani Matala-aho)
Abstract: If $\alpha$ is transcendental, then $P(\alpha)\ne 0$ for all polynomials $P$ with integer coefficients. The transcendence measure tells how far from zero these values of polynomials must be (at least). During my talk, I will first give a sketch of the current knowledge about the transcendence measure of $e$, and I will also briefly explain how these results can be obtained. I will then move to explaining how the transcendence measure can be generalised, and what is known and what is believed about the generalised transcendence measure.

20/05/15 4.00 Samuele Anni (Warwick)
Title: Residual modular Galois representations: images and applications.
Abstract: Let l be a prime number. To any mod l modular form, which is an eigenform for all Hecke operators, it is associated a 2-dimensional residual representation of the absolute Galois group of the rationals. Two different mod l modular forms can give rise to the same Galois representation. Analogously, a residual modular Galois representation can arise as twist of a representation of lower conductor. In this talk, after a brief introduction on residual modular Galois representations and mod l modular forms, I will address these problems and outline an algorithm for computing the image of such representations. I will also describe two applications of this algorithm (both still work in progress): solving Diophantine equations and graphs of modular forms.

27/05/15 4.00 Engeniy Zorin (York)
Title: Diophantine properties of Mahler numbers
Abstract: In this talk, I will explain what the class of Mahler Numbers is, present their Diophantine approximation properties and their link with Computer Science.

03/06/15 4.00 Pär Kurlberg (KTH)
Title: Ergodicity for point scatterers on arithmetic tori
Abstract: The Seba billiard was introduced to study the transition between integrability and chaos in quantum systems. The model seem to exhibit intermediate level statistics (i.e., repulsion between nearby eigenvalues, though not as strong as predicted by random matrix theory), as well as Gaussian value distribution of eigenfunctions ("wave chaos"). We investigate the very closely related "toral point scatterer"-model, namely eigenfunctions of the Laplacian perturbed by a delta-potential, on arithmetic 2D-tori. For a full density subsequence of "new" eigenfunctions we prove decay of matrix coefficients associated with pure momentum observables. This, together with previous work by Rudnick-Ueberschaer, allows us to conclude that quantum ergodicity holds for the set of "new" eigenfunctions. In particular, almost all new eigenfunction are equidistributed in both the position and the momentum representation. Time permitting we will discuss some recent "scar" constructions (i.e., sequences of eigenfuntions that do not equidistribute.)

10/06/15 4.00 Jon Keating (Bristol)
Title: Divisor Sums in Function Fields
Abstract - I will review some classical problems in number theory concerning sums of the (generalised) divisor function. I will then explain how analogues of these problems in the function field setting can be resolved by expressing them in terms of matrix integrals.

17/06/15 4.00 Alex Gorodnik (Bristol)
Title: Randomness in geometry of numbers
Abstract: We discuss the problem of counting solutions of Diophantine inequalities. While a general asymptotic formula for the counting function has been established by W. Schmidt, finer statistical properties of this function are still not well understood. We investigate its limiting distribution and establish the central limit theorem in this context. This is a joint work with Anish Ghosh.

24/06/15 4.00 Daniel Fiorilli (University of Ottawa and Paris 7)
Title: Chebyshev's bias for elliptic curves over function fields
Abstract: Since Chebyshev's observation that there seems to be more primes of the form 4n+3 than of the form 4n+1, many other types of 'arithmetical biases' have been found. For example, such a bias appears in the count of points on reductions of a fixed elliptic curve E; this bias is mainly created by the analytic rank. In this talk we will discuss the analogous question for elliptic curves over function fields. We will first discuss the occurrence of extreme biases, which originate from very different source than in the number field case. Secondly, we will discuss what happens to a 'typical curve', and discuss results of linear independence of the zeros of the associated L-functions. This is joint work with Byungchul Cha and Florent Jouve.

In Spring 2015, the seminar was hosted by UCL and organised by Sarah Zerbes.

14/01/15 Lucio Guerberoff (UCL)
Title: Shimura varieties and complex conjugation.

Abstract: Work of Shimura, Langlands, Milne-Shih, and, more recently, Taylor, have examined the action of complex conjugation on Shimura varieties. We study this topic from the viewpoint of the general theory of these varieties. In this talk, we will concentrate on certain Shimura varieties whose reflex field E is a CM field and stablish descent of these to the maximal totally real subfield E^+. We will discuss a strategy to construct these descent data in more generality, and mention possible applications. This work is joint with Don Blasius (UCLA) and remains in progress.

21/01/15 John Coates (Cambridge)
Title: Quadratic twists of elliptic curves.

Abstract: I will discuss joint work with Y. Li, Y. Tian and S. Zhai, which generalizes, for a wide class of elliptic curves defined over Q, the celebrated classical lemma of Birch and Heegner about quadratic twists with prime discriminants, to quadratic twists by discriminants having any prescribed number of prime factors. In addition, we prove stronger results for the family of quadratic twists of the modular elliptic curve X0(49), including showing that there is a large class of explicit quadratic twists whose complex L-series does not vanish at s = 1, and for which the full Birch-Swinnerton-Dyer conjecture is valid.

28/01/15 Fabrizio Andreatta (Milan)
Title: The eigencurve and its characteristic p special fibre.

Abstract: In the first part I will recall the approach I, Adrian Iovita, Glenn Stevens and independently Vincent Pilloni have developed in order to define families of overconvergent ellitpic modular forms as sections of suitable line bundles. I will then report on work in progress with Iovita and Pilloni which allows to extend those constructions to the boundary of the weight space, providing characteristic p Banach modules with a compact operator. This confirms an expectation of Robert Coleman.

4/2/15 Marc Masdeu (Warwick)
Title: Darmon points for number fields of mixed signature.

Abstract: Fifteen years ago Henri Darmon introduced a construction of p-adic points on elliptic curves. These points were conjectured to be algebraic and to behave much like Heegner points, although so far a proof remains inaccessible. Other constructions emerged in the subsequent years, thanks to work of himself and many others. All of these constructions are local (either non-archimedean like the original one, or archimedean), and so far none of these are proven to yield algebraic points, although there is extensive numerical evidence. In this talk I will present joint work with Xavier Guitart and Haluk Sengun, in which we propose a framework that includes all the above constructions as particular cases, and which allows us to extend the construction of local points to elliptic curves defined over arbitrary number fields. As a by-product, we provide an explicit (though conjectural) construction of the (isogeny class of the) elliptic curve attached to an automorphic form for GL2.

11/2/15 Andrew Booker (Bristol)
Title: L-functions as distributions.

Abstract: In 1989, Selberg defined what came to be known as the "Selberg class" of L-functions, giving rise to a new subfield of analytic number theory in the intervening quarter century. Despite its popularity, a few things have always bugged me about the definition of the Selberg class. I will discuss these nitpicks and describe some modest attempts at overcoming them, with new applications.

18/2/15 Pierre Colmez (Paris VI)
Title: p-adic local Langlands in weight 1.

Abstract: I will explain how one can recover the classical Langlands correspondence for GL(2,Qp) from the p-adic in weight one, and try to explain the relation with a conjecture of Breuil and Strauch.

Also 18/2/15 Wieslawa Niziol (ENS Lyon)
Title: Syntomic cohomology and p-adic nearby cycles.

Abstract: I will describe how the theory of (phi-Gamma) modules allows to relate p-adic nearby cycles and syntomic cohomology sheaves. This is a joint work with Pierre Colmez.

25/2/15 Levent Alpoge (Cambridge)
Title: The average elliptic curve has few integral points.

Abstract: It is a theorem of Siegel that the Weierstrass model y^2 = x^3 + Ax + B of an elliptic curve has finitely many integral points. A "random" such curve should have no points at all. I will show that the average number of integral points on such curves (ordered by height) is bounded --- in fact, by 67. The methods combine a Mumford-type gap principle, LP bounds in sphere packing, and results in Diophantine approximation. The same result also holds (though I have not computed an explicit constant) for the families y^2 = x^3 + Ax, y^2 = x^3 + B, and y^2 = x^3 - n^2 x.

4/3/15 Carlos de la Mora (UEA)
Title: On a generalisation of local coefficients

Abstract: Some of the most important problems in number theory are the Langlands Con- jectures. These conjectures are very general and even the simplest' cases have very strong applications. The statement of the Langlands conjectures depends on the existence of certain local factors that ought to be attached to smooth representations of reductive groups over local elds. These local factors have not yet been defined in general, but in the case where we have a quasi-split group and a generic representation, they have been defined by the so called Langlands-Shahidi method. One of the main ingredients in the Langlands-Shahidi method is the definition of the local coefficients. The main intent of our research is to find a family of functions that behave like eigenvectors for the intertwining operators such that the eigenvalues' are the local coefficients. We have developed this construction with the vain of obtaining a generalization of local coefficients of certain representations of an arbitrary reductive group over non-archimedean local field.

11/3/15 Andreas Langer (Exeter)
Title: Grothendieck-Messing deformation theory for varieties of K3-type.

Abstract: For an artinian local ring R with perfect residue field we define higher displays over the small Witt ring. The second crystalline cohomology of a variety of K3-type X (for example the Hilbert schemes of zero-dimensional subschemes of a K3-surface) is equipped with the additional structure of a 2-display. Then we extend the Grothendieck-Messing lifting theory from p-divisible groups to such varieties: The deformations of X over a nilpotent pd-thickening correspond uniquely to selfdual liftings of the associated Hodge-filtration. For the proof we give an algebraic definition of the Beauville-Bogomolov-Form on the second de Rham cohomology of X and show that for ordinary varieties the deformations of X correspond uniquely to selfdual deformations of the 2-display endowed with its Beauville-Bogomolov form. This is joint work with Thomas Zink.

Title: D-modules on rigid analytic spaces

Abstract: In this talk I will discuss some recent work with Konstantin Ardakov that seeks to develop a framework for a theory of D-modules for rigid analytic spaces in order to better understand the locally analytic representation theory of p-adic analytic groups. To be more precise, we define and study a canonical completion of the sheaf of classical differential operators on a rigid analytic space (in the sense of Tate) that may be viewed as a quantization of the sheaf of functions on the rigid analytic cotangent bundle. We introduce what we call 'co-admissible modules' for this sheaf of non-commutative rings. When the rigid analytic space is the flag variety of a split semisimple p-adic analytic group then there is an equivalence of categories between the category of "co-admissible D-modules" and co-admissible modules (in the sense of Schneider--Teitelbaum) of a certain canonical completion of the enveloping algebra of the associated Lie algebra with trivial infinitesimal central character. Much of this is written up in recent preprints that can be found on the ArXiv; some of it is not yet written up.

In Autumn 2014, the seminar was hosted by Imperial College and organised by Kevin Buzzard.

08/10/14 Jack Thorne (Cambridge University)
Title: Arithmetic of plane quartic curves.

Abstract: Bhargava, Gross and Wang have studied the group J(k)/2J(k), J the Jacobian of a hyperelliptic curve over a field k, using representation theory and invariant theory. In this talk, I will outline a similar program for smooth plane quartic curves (which are non-hyperelliptic of genus 3) with a marked rational point.

15/10/14 Rene Pannekoek (Imperial)
Title: Explicit unbounded ranks of Jacobians in towers of function fields.

Abstract: This is joint work with Lisa Berger, Chris Hall, Jennifer Park, Rachel Pries, Shahed Sharif, Alice Silverberg, Douglas Ulmer. For each prime power q and each odd prime r not dividing q, we define a curve C of genus r-1 over F_q(t). We give explicit generators for a subgroup of (Jac(C))(K), where K runs through a tower of extensions of F_q(t), and prove that the rank of this subgroup grows linearly with [K:F_q(t)]. By constructing a proper regular model, we also prove that the subgroup is actually of finite index.

22/10/14 Tony Scholl (Cambridge)
Title: Plectic cohomology of Shimura varieties

Abstract: I will discuss a new, largely conjectural, cohomology theory for a certain class of Shimura varieties, and explain some of its arithmetic applications. This is joint work with Jan Nekovar.

29/10/14 Olivier Taibi (Imperial)
Title: Computing dimensions of spaces of automorphic/modular forms for classical groups using the trace formula

Abstract: Let G be a Chevalley group which is symplectic or special orthogonal. I will explain how to explicitly compute the geometric side of Arthur's trace formula for a function on G(AA) which is a stable pseudo-coefficient of discrete series at the real place and the unit of the unramified Hecke algebra at every finite place. Arthur's recent endoscopic classification of the discrete automorphic spectrum of G allows to analyse the spectral side in detail. For example one can deduce dimension formulae for spaces of vector-valued Siegel modular forms in level one. The computer achieves these computations at least up to genus 7.

5/11/14 James Maynard (Oxford)
Title: Large gaps between primes

Abstract: A 1938 result of Rankin shows that there are consecutive primes less than $x$ whose difference is $\gg (\log{x})(\log\log{x})(\log\log\log\log{x})/(\log\log\log{x})^2$. Over the past 75 years, improvements have only been in the implied constant. We will show how one can use recent progress on small gaps between primes to quantitatively improve this bound. A similar improvement was found independently by Ford, Green, Konyagin and Tao using different techniques.

Monday 10/11/14: London-Paris number theory seminar (in Paris).

12/11/14 Samir Siksek
Title: Modularity of elliptic curves over totally real fields

Abstract: We combine the latest advances in modularity lifting with a 3-5-7 modularity switching argument to deduce modularity of 'most' elliptic curves over totally real fields. In particular, we show that all elliptic curves over real quadratic fields are modular. This talk is based on joint work with Bao Le Hung (Harvard) and Nuno Freitas (Bonn).

19/11/14 Chris Blake (Cambridge)
Title: The F-plectic Taniyama group

Abstract: Scholl and Nekovar conjecture that, in the presence of real multiplication by a totally real field F, certain motives should carry a canonical ("F-plectic") structure. I will talk about a first step towards showing the existence of such "canonical F-plectic models" for Shimura varieties attached to groups of the form G = Res_{F/Q} H. More precisely I will explain how Langlands' construction of the Taniyama group (which played a key role in proving the existence of canonical models for Shimura varieties) can be generalised to the plectic setting.

26/11/14 Judith Ludwig (Imperial)

Abstract: In this talk we will study an example of p-adic Langlands functoriality: Let B be a definite quaternion algebra over the rationals, G the algebraic group defined by the units in B and H the subgroup of G of norm one elements. Then the classical transfer of automorphic representations from G to H is well understood thanks to Labesse and Langlands, who proved formulas for the multiplicity of irreducible admissible representations of H(adeles) in the discrete automorphic spectrum. In this talk we will prove a p-adic version of this transfer. More precisely we will extend the classical transfer to p-adic families of automorphic forms as parametrized by eigenvarieties. We will prove the p-adic transfer by constructing a morphism between eigenvarieties, which agrees with the classical transfer on points corresponding to classical automorphic representations.

3/12/14 David Helm (Imperial)
Title: Maximal tori and the integral Bernstein center for GL_n

Abstract: The integral Bernstein center is an algebra that acts naturally on all smooth \ell-adic representations of GL_n(F), for F a p-adic field. If one inverts \ell this algebra has a nice description due to Bernstein and Deligne, but in the integral context it is much more complicated. We show that this algebra has a clean description in terms of the maximal tori of GL_n/F, and use this description to relate the Bernstein center to the deformation theory of Galois representations via local Langlands.

10/12/14 Roger Heath-Brown (Oxford)
Title: Primes of the form a^2+p^4

Absract: We show that there are infinitely many primes of the form a^2+p^4, with p prime, thereby sharpening a result of Friedlander and Iwaniec in which p was not restricted to be prime. The talk will illustrate the general strategy for such problems, and highlight a superficially surprising lemma on the equidistribution of primes in congruence classes.

In Summer 2014, the seminar was hosted by Kings College and organised by Igor Wigman and Mahesh Kakde.

23/04/14 Jack Lamplugh (Cambridge University)
Title: Class numbers in some non-cyclotomic $\mathbb{Z}_p$-extensions.
Abstract: The class numbers of the finite layers of the cyclotomic $\mathbb{Z}_p$-extensions of the rationals have been the subject of much study. By studying the absolute height of cyclotomic units, Horie effectively proved that 100% of prime numbers $q$ never divide the class numbers of the finite layers of the cyclotomic $\mathbb{Z}_p$-extension of the rationals. I will talk about how similar methods can be applied to $\mathbb{Z}_p$-extensions of an imaginary quadratic field $K$ which are unramified outside of a split prime above $p$.

30/04/14 Yiannis Petridis (UCL)
Title: On the hyperbolic lattice-point problem
Abstract:

07/05/14 Avner Ash (Boston College)
Title: Reducible Galois representations and homology of $GL(n,\mathbb{Z})$.
Abstract: In joint work with Darrin Doud, I have connected some reducible 3-dimensional representations of the absolute Galois group of $\mathbb{Q}$ with homology Hecke eigenclasses for congruence subgroups of $GL(3,\mathbb{Z})$. I will explain what we have done and how it fits into the picture of generalized reciprocity laws for $GL(n,\mathbb{Z})$.

14/05/14 David Hansen (Institut de mathématiques de Jussieu)
Title: Eigenvarieties beyond the case of discrete series
Abstract: Given a connected reductive group $G$ over a number field $F$, I'll explain a construction of eigenvarieties parametrizing "Betti-cohomological overconvergent $p$-adic modular forms" on $G$ under a very mild hypothesis (namely that $G$ splits over $F_v$ for each $v|p$). This construction generalizes previous works of a great number of authors, which have largely focused on case where $G(F \otimes R)$ has a discrete series. Beyond the setting of discrete series, the qualitative geometric properties of eigenvarieties change drastically. I'll discuss the known and expected properties of these spaces, and their (largely conjectural) relationship with Galois representations.

21/05/14 4.00 Gergely Zabradi (Eötvös Lorànd University, Budapest)
Title: Algebraic functional equations and completely faithful Selmer groups
Abstract: Let $E$ be an elliptic curve---defined over a number field $K$---without complex multiplication and with good ordinary reduction at all the primes above a rational prime $p\geq 5$. We construct a pairing on the dual $p^\infty$-Selmer group of $E$ over any strongly admissible $p$-adic Lie extension $K_{\infty}/K$ under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group. This gives an algebraic functional equation of the conjectured $p$-adic $L$-function. As an application we construct completely faithful Selmer groups in case the $p$-adic Lie extension is obtained by adjoining the $p$-power division points of another non-CM elliptic curve $A$. This is joint work with T. Backhausz.

28/05/14 4.00 Pankaj Vishe (University of York)
Title: Inhomogeneous multiplicative Littlewood conjecture and logarithmic savings
Abstract: We use the dynamics on $SL(3,\mathbb{R})/SL(3,\mathbb{Z})$ to get logarithmic savings in the inhomogeneous multiplicative Littlewood setting. This is a joint work with Alex Gorodnik.

04/06/14 4.00 Dzmitry Badziahin (Durham University)
Title: On potential counterexamples to mixed Littlewood conjecture.
Abstract: Mixed Littlewood conjecture proposed by de Mathan and Teulie in 2004 states that for every real number $x$ one has $$\liminf_{q\to\infty} q\cdot |q|_D\cdot ||qx|| = 0.$$ where $|*|_D$ is a s called pseudo norm which generalises the standard $p$-adic norm. In the talk we'll consider the set $\mad$ of potential counterexamples to this conjecture. Thanks to the results of Einsiedler and Kleinbock we already know that the Haudorff dimension of $\mad$ is zero, so this set is very tiny. During the talk we'll see that the continued fraction expansion of every element in $\mad$ should satisfy some quite restrictive conditions. As one of them we'll see that for these expansions, considered as infinite words, the complexity function can neither grow too fast nor too slow.

Monday 9/06/14: London-Paris number theory seminar: details are here.

11/06/14 Minhyong Kim (Oxford University)
Title: "Non-abelian reciprocity laws and Diophantine geometry"

18/06/14 Andrew Wiles (Oxford)
Title: On class groups of imaginary quadratic fields

25/06/14 Zeev Rudnick (Tel-Aviv University)
Title: "Some problems in analytic number theory for polynomials over a finite field".
Abstract: The lecture explores several problems of analytic number theory in the context of function fields over a finite field, where they can be approached by methods different than those of traditional analytic number theory. The resulting theorems can be used to check existing conjectures over the integers, and to generate new ones. Among the problems discussed are: Counting primes in short intervals and in arithmetic progressions; Chowla's conjecture on the autocorrelation of the Mobius function; the distribution of squarefree numbers. The tools used are from analytic number theory, algebraic geometry, field arithmetic and beyond.

In Spring 2014, the seminar was hosted by UCL, and was organized by Sarah Zerbes.

15/01/14 Alex Bartel (Warwick)
Title: The Cohen-Lenstra heuristic revisited

22/01/14 James Newton (Cambridge)
Title: Local-global compatibility for Galois representations associated to Hilbert modular forms of low weight

29/01/14 Chris Williams (Warwick)
Title: Overconvergent modular symbols over imaginary quadratic fields

05/02/14 Victor Rotger (Univ. Polytècnica de Catalunya)
Title: The Birch and Swinnerton-Dyer conjecture for non-abelian twists of elliptic curves.

12/02/14 At 2:30 in 707: Masato Kurihara (Keio University)
Title: Higher Fitting ideals of arithmetic objects

12/02/14 At 4:00 in 707: Thanasis Bouganis (Durham)
Title: p-adic measures for Hermitian modular forms and the Rankin-Selberg method

19/02/14 Matthew Morrow (Nottingham)
Title: Deformation of algebraic cycles in characteristic p

26/02/14 Guido Kings (Regensburg)
Title: Eisenstein classes in syntomic cohomology and applications

05/03/14 Malte Witte (Heidelberg)
Title: Grothendieck trace formulas and Iwasawa main conjectures in characteristic p

12/03/14 Chris Lazda (Imperial)
Title: Rigid rational homotopy theory and mixedness

19/03/14 Peter Sarnak (Princeton) ( NB: talk will be in room 500)
Title: Families of L-functions and their symmetry

26/03/14 Jenny Cooley (Warwick)
Title: Mordell-Weil point generation on cubic surfaces over finite fields

In Autumn 2013, the seminar was hosted by Imperial College, and organized by Alexei Skorobogatov.

2/10/13 Toby Gee (Imperial)
Title: The p-adic local Langlands correspondence beyond GL_2(Q_p).

9/10/13 Luis Garcia (Imperial)
Title: Periods of modular forms on Shimura curves and singular theta lifts

16/10/13 Tim Dokchitser (Bristol)
Title: L-functions of curves.

23/10/13 Rebecca Bellovin (Imperial)
Title: p-adic Hodge theory in rigid analytic families

30/10/13 Martin Taylor (Oxford)
Title: Second Chern class and SK1

6/11/13 René Pannekoek (Imperial)
Title: The Brauer-Manin obstruction and ranks of quadratic twists

13/11/13 David Helm (Imperial)
Title: A derived local Langlands correspondence for GL_n

20/11/13 Bob Hough (Cambridge)
Title: The minimum modulus of a covering system is bounded.

27/11/13 Brandon Levin (IAS)
Title: G-valued flat deformations and local models.

4/12/13 Martin Orr (UCL)
Title: Unlikely intersections in abelian varieties and Shimura varieties.

11/12/13 Lilian Matthiesen (Orsay)
Title: Generalised Fourier coefficients of multiplicative functions and applications.

In Summer 2013, the seminar was hosted by Kings College, and organized by Mahesh Kakde.

24/4/13 2:30: P. Cassou-Noguès (Bordeaux)
Title: On simple rational polynomials.
24/4/13 4:00 (same day): Ph. Cassou-Noguès (Bordeaux)
Title: Cohomological invariants of quadratic forms

1/5/13 Igor Wigman (KCL)
Title: Nodal length fluctuations for arithmetic random waves.

Title: Galois equivariance of L-values and the Birch-Swinnerton-Dyer conjecture

15/5/13 Sanju Velani (York)
Title: Multiplicative and Inhomogeneous Diophantine Approximation

22/5/13 Stefano Morra (Montpellier)
The action of compact subgroups on some p-modular automorphic representations of GL(2,Q_p)

29/5/13 Dinesh Thakur (Arizona)
Title: Higher multiplicities in number theory

3-4/6/13: London-Paris Number Theory Seminar (in London). Speakers: F. Charles (Rennes), Y. Harpaz (Nijmegen), J-L. Colliot-Thelene (Orsay), O. Wittenberg (ENS), D. Testa (Warwick)

5/6/13 P. Xu (UEA)
Title: On the irreducible mod p representations of unramfied U(2,1)

12/6/13 Luis Dieulefait (Barcelona)
New results on Langlands functoriality: non-solvable base change, symmetric powers and tensor products.

19/6/13 Christopher Daw (UCL)
Title: Degrees of strongly special subvarieties and the André-Oort conjecture.

26/6/13 Ashay Burungale
Title: P-rigidity and Iwasawa mu-invariants.

In Spring 2013, the seminar was hosted by UCL. Sarah Zerbes was the organizer.

16/01 Speaker: David Loeffler (Warwick)
Title: Euler systems for the Rankin-Selberg convolution of modular forms.

23/01 Speaker: Nicola Mazzari (Paris 7)
Title: On the syntomic regulator

30/01 Speaker: Laurent Berger (ENS Lyon)
Title: The p-adic local Langlands correspondence and Lubin-Tate groups 06/02 Speaker: Judith Ludwig (Imperial)
Title: p-adic functoriality for inner forms of unitary groups

13/02 Speaker: Gebhard Boeckle (Heidelberg)
Title: Independence of ell-adic Galois representations over finitely generated fields

20/02 Speaker: John Coates (Cambridge)
Title: Congruent numbers.

27/02 Speaker: Amnon Besser (Ben Gurion University/Oxford)
Title: Toric regulators for varieties with totally degenerate reduction over p-adic fields

06/03 NO SEMINAR

13/03 Speaker: Massimo Bertolini (Milan)
Title: Kato's reciprocity law and a p-adic Beilinson formula

20/03 Speaker: Damiano Testa (Warwick)
Title: The Buechi K3 surface and its rational points

In Autumn 2012, the seminar was hosted by Imperial College, and was organized by Ambrus Pal.

03/10 Speaker: Andrei Yafaev (UCL)
Title: Hyperbolic Ax-Lindemann theorem in the cocompact case
Abstract: This is a joint work with E. Ullmo. The classical Ax-Lindemann theorem is a statement in transcendence theory of the exponential function. We prove an analogous statement for the map uniformising compact Shimura varieties.

10/10 Speaker: Nuno Freitas (Kings College London)
Title: Fermat-type equations of signature (13,13,p) via Hilbert cuspforms
Abstract: In this talk I am going to discuss how a modular approach via Hilbert cuspforms can be used to show that equations of the form x^13 +y^13 = Cz^p have no non-trivial primitive solutions (a,b,c) such that 13 does not divide c if p > 4992539. We will first relate a putative solution of the previous equation to the solution of another Diophantine equation with coefficients in the real quadratic extension of Q with discriminant 13. Then we attach Frey curves E over this field to solutions of the latter equation. Finally, by proving modularity of E and irreducibility of certain Galois representations attached to E we are able to apply a modular approach via Hilbert modular forms.

17/10 Speaker: Toby Gee (Imperial College London)
Title: Patching functors and the cohomology of Shimura curves
Abstract: I will explain recent joint work with Matthew Emerton and David Savitt, in which we relate the geometry of various tamely potentially Barsotti-Tate deformation rings for two-dimensional Galois representations to the integral structure of the cohomology of Shimura curves. As a consequence, we establish some conjectures of Breuil regarding this integral structure. The key technique is the Taylor-Wiles-Kisin patching argument, which, when combined with a new, geometric perspective on the Breuil-Mezard conjecture, forges a tight link between the structure of cohomology (a global automorphic invariant) and local deformation rings (local Galois-theoretic invariants).

Monday October 22nd: London-Paris Number Theory Seminar. Speakers: Sarah Zerbes (UCL), David Vauclair (Caen), David Burns (Kings College London).

24/10 Speaker: Tom Fisher (Cambridge)
Title: Visibility of Tate-Shafarevich groups in abelian surfaces and abelian threefolds
Abstract: Let E be an elliptic curve over the rationals. The Tate-Shafarevich group of E measures the failure of the Hasse Principle for principal homogeneous spaces under E. Mazur suggested that one should attempt to visualize'' elements of this group as cosets of E inside some larger abelian variety A. I will discuss what is known about the visibility dimension (i.e. the least possible dimension of A) and give some examples of elements of order 7 that are visible in abelian surfaces and abelian threefolds.

31/10 Speaker: Nicolas Ojeda Bar (Cambridge)
Title: Towards the cohomological construction of Breuil-Kisin modules.
Abstract: We will present an approach to the construction of Breuil-Kisin modules using crystalline cohomology. Along the way, we will define a new PD-base for crystalline cohomology that improves in some respects upon the ring S used by Breuil and Faltings in their appraoch to integral p-adic Hodge theory.

7/11 Speaker: Mehmet Haluk Sengun (Warwick)
Title: Cohomology of Bianchi Groups and Arithmetic
Abstract: Bianchi groups are groups of the form SL(2,R) where R is the ring of an imaginary quadratic field. They arise naturally in the study of hyperbolic 3-manifolds and of certain generalisations of the classical modular forms (called Bianchi modular forms) for which they assume the role of the classical modular group SL(2,Z). After giving the necessary background, I will start with a discussion of the problem of understanding the behaviour of the dimensions of the cohomology of Bianchi groups and their congruence subgroups. Next, I will focus on the amount of the torsion that one encounters in the cohomology . Finally, I will discuss the arithmetic significance of these torsion classes.

14/11 Speaker: Abhishek Saha (Bristol)
Title: Quantum Unique Ergodicity for holomorphic newforms
Abstract: Let f be a classical holomorphic newform of level q and even weight k. I will describe recent joint work with Paul Nelson and Ameya Pitale where we prove that the pushforward to the full level modular curve of the mass of f equidistributes as qk goes to infinity. This generalizes previous work by Holowinsky-Soundararajan (the case q=1, k-> infinity) and Nelson (the case qk -> infinity over squarefree integers q). Thus we settle the holomorphic quantum unique ergodicity conjecture in all aspects (for classical modular forms of trivial nebentypus). A potentially surprising aspect of our work is that we obtain a power savings in the rate of equidistribution as q becomes sufficiently powerful'' (far away from being squarefree), and in particular in the depth aspect'' as q traverses the powers of a fixed prime.

21/11 Speaker: Alexei Skorobogatov (Imperial College London)
Title: Rational points on conic and quadric bundles with many degenerate fibres
Abstract: This is a talk on the joint work with Tim Browning and Lilian Matthiesen. We apply additive combinatorics of Green and Tao to prove that the Brauer-Manin obstruction controls weak approximation for rational points on pencils of conics or quadrics over Q, provided that all singular fibres are defined over Q.

28/11 Speaker: Paul Ziegler (ETH Zurich)
Title: Mordell-Lang in positive characteristic
Abstract: I will talk about the Mordell-Lang conjecture in positive characteristic and sketch a new algebro-geometric proof for the case that the subgroup involved is finitely generated.

05/12 Speaker: Bruno Angles (Caen)
Title: The Herbrand-Ribet theorem for function fields revisited
Abstract: We will present a new proof of Taelman's Herbrand-Ribet Theorem based on an equivariant class number formula for cyclotomic function fields recently obtained in a joint work with L. Taelman. If we have time, we will also show how this equivariant formula implies that a conjecture "à la Vandiver" made by G. Anderson in 1996 is false.

12/12 Speaker: Frank Neumann (Leicester)
Title: Etale Homotopy of Deligne-Mumford stacks.
Abstract: We will give an overview of etale homotopy theory a la Artin-Mazur of Deligne-Mumford stacks and discuss several examples including moduli stacks of algebraic curves and principally polarised abelian varieties.

In Summer 2012, the seminar was hosted by KCL, and was organised by Fred Diamond.

### LNTS Schedule

25 April: Werner Mueller, Bonn
"Applications of the trace formula to spectral theory of automorphic forms."

2 May: Stefano Vigni, King's
"Darmon points on Shimura curves and abelian varieties"

9 May: David Savitt, Arizona
"The Buzzard-Diamond-Jarvis conjecture for unitary groups"

16 May: Riccardo Brasca, Milan
"p-adic modular forms of non-integral weight over Shimura curves"

23 May: Konstantin Ardakov, QMUL
"Quillen's Lemma for affinoid enveloping algebras"

30 May: London-Paris Number Theory Seminar. Speakers: Joel Rivat (Luminy), Gerald Tenenbaum (Nancy), Adam Harper (Cambridge).

6 June: Jens Marklof, Bristol
"Circulant graphs, Frobenius numbers and ergodic theory"

13 June: William Conley, UEA
"Inertial types for automorphic representations"

20 June: Alex Paulin, Nottingham

27 June: Christian Johansson, Imperial
"Classicality for overconvergent automorphic forms on some quaternionic Shimura varieties"

In Spring 2012, the seminar was hosted by UCL, and was organised by Andrei Yafaev.

### LNTS Schedule

Jan 18: Florence Gillibert (Bordeaux and MPI Bonn)
Title: Rational points on Atkin-Lehner quotients of Shimura curves

Jan 25 Roger Heath-Brown (Oxford)

Feb 1st: Jan Nekovar (Paris)
Title: The method of Bertolini and Darmon

Feb 8: Richard Hill (UCL)

Feb 15 Paul-James White (Oxford)

Feb 22nd bonus talk at 2:30: Go Yamashita (Toyota)

Feb 29th Jonathan Pila (Oxford)
Title: Some unlikely intersections beyond Andre-Oort.

March 7th Rene Pannekoek (Leiden)
Title: On the p-adic density of the rational points on K3 surfaces.

March 14th Xavier Caruso
Title: An algorithm to compute the semi-simplification modulo p of a semi-stable representation

March 21 Anish Ghosh (Norwich)
Title: Effective density of rational points on homogeneous varieties

In Autumn 2011, the seminar was hosted by Imperial, and organised by Ambrus Pal.

### LNTS Schedule

DATE: 05/10/11

Speaker: Kevin Buzzard (Imperial College)

Title: Reduction of crystalline representations

Abstract: 2-dimensional crystalline Galois representations of the absolute Galois group of Q_p can be completely determined by linear algebra data. One can also classify 2-dimensional mod p Galois representations easily. This leads us to the following question: given a piece of linear algebra data, what is the reduction of the corresponding p-adic Galois representation? This question turns out to be a little subtle. I'll survey the state of the art and talk about the most recent method of attack -- the p-adic and mod p Langlands Program.

-----------------------------------

DATE: 12/10/11

Speaker: Ivan Tomasic (Queen Mary)

Title: Twisted Galois stratification

Abstract: The aim is to present the development of difference algebraic geometry and its applications to counting solutions of difference polynomial equations over fields with powers of Frobenius. We prove a twisted version of Chebotarev's theorem for a Galois covering of difference schemes, and use it to deduce an important direct image theorem: the image of a "twisted Galois formula" by a morphism of difference schemes is again a twisted Galois formula.

-----------------------------------

DATE: 19/10/11

Speaker: Jamshid Derakhshan (Oxford)

Title: Model theory of the adeles and connections to number theory

Abstract: This is joint work with Angus Macintyre. Model theory studies definable subsets of a structure in a specific language. For many important structures, definable sets turn out to have a rich geometry in a natural language. Once the family of definable sets has a 'a direct image theorem', the structure of definable sets becomes transparent. This usually implies decidability, but there are also applications to geometry and arithmetic; and structures enjoying such properties can be thought of as 'tame' in some sense.

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DATE: 26/10/11

Speaker: Francois Loeser (Jussieu)

Title: Motivic height zeta functions and the Poisson formula

Abstract: Recently, Chambert-Loir and Tschinkel obtained asymptotic formulas for the number of integral points of bounded height on partial equivariant compactifications of vector groups. In this talk we shall present a motivic version of these results. We show the rationality of the corresponding motivic height zeta functions and determine its leading pole and the residue. Our approach relies on "motivic harmonic analysis". In particular a motivic Poisson formula due to Hrushovski and Kazhdan plays a key role. This is joint work with Antoine Chambert-Loir.

-----------------------------------

DATE: 02/11/11

Speaker: Alena Pirutka (Strasbourg)

Title: On some aspects of unramified cohomology

Abstract: During this talk, I would like to explain different interactions of unramified cohomology groups with other problems, such as the study of Chow groups or some local-global principles.

-----------------------------------

DATE: 09/11/11

Speaker: Toby Gee (Imperial College)

Title: p-adic Hodge-theoretic properties of etale cohomology with mod p coefficients, and the cohomology of Shimura varieties

Abstract: I will discuss some new results about the etale cohomology of varieties over a number field or a p-adic field with coefficients in a field of characteristic p, and (if time permits) give some applications to the cohomology of unitary Shimura varieties. (Joint with Matthew Emerton.)

-----------------------------------

DATE: 16/11/11

Speaker: Bruno Angles (Caen)

Title: On the class group module of a Drinfel'd module

Abstract: Recently L. Taelman has constructed a finite F_q[T]-module attached to a Drinfel'd module which is an analogue of the ideal class group of a number field. Taelman has proved an analytic class number formula for these modules and an analogue of Ribet's Theorem. In this talk, we will consider an analogue of the Kummer-Vandiver problem for these modules and we will present examples which give a negative answer to this problem. This talk is based on a joint work with L. Taelman.

-----------------------------------

DATE: 23/11/11

Speaker: Teruyoshi Yoshida (Cambridge)

Title: The Hecke action on the weight spectral sequences

Abstract: We review the question of making algebraic correspondence act on the weight spectral sequence for l-adic cohomology of semistable schemes. To do this we need some intersection theory and cycle classes on regular schemes over the ring of integers. This approach works for the unitary Shimura varieties considered by Harris-Taylor/Shin (as the Hecke correspondences are finite and flat).

-----------------------------------

DATE: 30/11/11

Speaker: Peter Scholze (Bonn)

Title: On the cohomology of compact unitary group Shimura varieties at ramified split places

Abstract: Generalizing our previous methods, we give a description of the cohomology of Shimura varieties for which the reductive group G is locally at p a product of general linear groups, allowing arbitrary signature at infinity and arbitrary ramification at p. As applications, we give a complete description of the semisimple local Hasse-Weil zeta function in terms of automorphic L-functions in nonendoscopic cases, and reconstruct the l-adic Galois representations attached to RACSD cuspidal automorphic representations, using endoscopic cases. This is joint work with Sug Woo Shin.

-----------------------------------

DATE: 07/12/11

Speaker: Ambrus Pal (Imperial College)

Title: Around de Jongâ€™s conjecture

Abstract: I will talk about one item of my current work in progress, which gives a new proof of de Jongâ€™s conjecture in the rank two case, and a closely related analogue of Serreâ€™s conjecture for function fields.

-----------------------------------

DATE: 14/12/11

Speaker: Andreas Langer (Exeter)

Title: An integral structure on rigid cohomology

Abstract: For a quasiprojective smooth variety over a perfect field k of char p we introduce an overconvergent de Rham-Witt complex by imposing a growth condition on the de Rham-Witt complex of Deligne-Illusie using Gauus norms and prove that its hypercohomology defines an integral strcuture on rigid cohomology, i.e. its image in rigid cohomology is a canonical lattice. As a corollary we obtain that the integral Monsky-Washnitzer cohomology (considered before inverting p) of a smooth k-algebra is of finite type modulo torsion. This is joint work with Thomas Zink.

In Summer 2011, the seminar was hosted by KCL, and organised by David Solomon.

### LNTS Schedule

DATE: 27/4/11

SPEAKER: Alan Lauder (Oxford)

TITLE: "Computations with classical and overconvergent modular forms"

ABSTRACT: We present p-adic algorithms for computing Hecke polynomials and Hecke eigenforms associated to spaces of classical modular forms using the theory of overconvergent modular forms.

-----------------------------------

DATE: 4/5/11

SPEAKER: Christian Elsholtz (Graz University of Technology)

TITLE: "Sums of Unit Fractions"

ABSTRACT: We study the diophantine equation

$\frac{m}{n}= \frac{1}{x_1}+ \ldots +\frac{1}{x_k}$

(with positive integers $m,n,x_i$) and give a survey of several questions in this area.

-----------------------------------

DATE: 11/5/11

SPEAKER: Victor Rotger (Universitat Politecnica de Catalunya)

TITLE: "Sato-Tate distributions and Galois endomorphism modules"

ABSTRACT: Let A be an abelian variety of dimension g over a number field k. For any prime p of k at which A has good reduction let L_p(A,T) be the characteristic polynomial of a Frobenius at p, unitarized in such way that its trace a_p(A) lies in [-2g,2g].

One wonders how these traces are distributed in this interval as p varies. When g=1, the classical results of Deuring and the recent breakthrough of Clozel, Harris, Shepherd-Barron and Taylor on the conjecture of Sato-Tate offer a pretty satisfactory answer to this question. In a recent beautiful paper, Kedlaya and Sutherland explore in detail next case g=2, proposing a quite precise conjectural description of the fauna of distributions that can arise.

In this talk we shall review their work, offering a different point of view and completing their picture at several points. This is a joint project with F. Fite, K. Kedlaya and A. Sutherland.

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DATE: 18/5/11

SPEAKER: Gunther Cornelissen (Utrecht)

TITLE: "What determines a number field? - A view from quantum statistical mechanics"

ABSTRACT: I will start with an overview of the history of (not) reconstructing number field isomorphism from equality/isomorphism of invariants such as zeta functions, adele rings and abelian/absolute Galois groups. Then I will discuss joint work with Matilde Marcolli that reconstructs isomorphism of global fields from isomorphism of associated quantum statistical mechanical systems (which are certain dynamical systems derived from class field theory), and how this implies that abelian L-series determine the isomorphism type of a global field.

-----------------------------------

DATE: 25/5/11

SPEAKER: Fred Diamond (KCL)

TITLE: TBA

ABSTRACT: TBA

-----------------------------------

DATE: 1/6/11

-----------------------------------

DATE: 8/6/11

SPEAKER: Burt Totaro (Cambridge)

TITLE: "Pseudo-abelian varieties"

ABSTRACT: Chevalley's theorem states that every smooth connected algebraic group over a perfect field is an extension of an abelian variety by a smooth connected affine group. That fails when the base field is not perfect. We define a class of smooth connected groups over an arbitrary field k called pseudo-abelian varieties; for k perfect, these are simply abelian varieties. The definition is arranged so that every smooth connected group over a field is an extension of a pseudo-abelian variety by a smooth connected affine group, in a unique way.

We work out much of the structure of pseudo-abelian varieties. These groups are closely related to unipotent groups in characteristic p and to pseudo-reductive groups as studied by Tits and Conrad-Gabber-Prasad. Many properties of abelian varieties such as the Mordell-Weil theorem extend to pseudo-abelian varieties.

-----------------------------------

DATE: 15/6/11

SPEAKER: Wojciech Gajda (Adam Mickiewicz University, Poznan)

TITLE: "Independence of \ell-adic representations over function fields and abelian varieties"

ABSTRACT: Let K be a finitely generated field extension of Q. We consider a family of \ell-adic representations (\ell varies through rational primes) of the absolute Galois group of K, attached to the \ell-adic cohomology of a fixed separated scheme of finite type over K.

We prove that the fields cut out from the algebraic closure of K by the kernels of the representations of the family are linearly disjoint over a finite extension of K. This gives a positive answer to a question formulated by Serre in 1991.

In addition (if time permits) we will also discuss some recent calculations of images of \ell-adic Galois representations attached to Tate modules of abelian varieties over number fields, and applications to geometry and to arithmetic.

-----------------------------------

DATE: 22/6/11

SPEAKER: Daniel Macias Castillo (KCL)

TITLE: "On Higher-Order Stickelberger-Type Theorems"

ABSTRACT: We discuss a conjecture concerning the annihilation, as Galois modules, of ideal class groups by explicit elements constructed from the values of higher order derivatives of Dirichlet L-functions. We describe evidence in support of this conjecture, including a full proof in several important cases.

## Spring 2011

This term, the seminar was held at Kings College and was organized by Andrei Yafaev.

#### The seminar program for Spring 2011

Jan 19: Laurent Fargues (Orsay, Paris-11) "Curves and vector bundles in p-adic Hodge theory"

Jan 26 Berhang Noohi (KCL) "Uniformization of orbifold Riemann surfaces"

Abstract: Riemann's uniformization theory says that there are exactly three simply connected Riemann surfaces: Riemann sphere, complex plane, and the hyperbolic plane. Moreover, the quotient of a free (discrete) group action on any of these will give rise to a Riemann surface, and every Riemann surface is obtained this way. Now, if we allow the action to be properly discontinuous but have fixed points, the quotient will be an orbifold Riemann surface. The question is whether all orbifold Riemann surfaces are obtained this way. The answer is no. In this talk we establish uniformization theory of orbifold Riemann surfaces. We show that there is an orbifold version of the spherical/Euclidean/hyperbolic trichotomy, and show how to classify orbifold Riemann surfaces by their uniformization type. There will be lots of examples in this talk! (This is joint work with Kai Behrend.)

Feb 2nd Lynne Walling (Bristol) "Quadratic forms, representation numbers, and Siegel-Eisenstein series."

Feb 9th Trevor Wooley (Bristol) "Vinogradov's mean value theorem, function fields, and moduli spaces"

Feb 16th Martin Bright (Warwick)

Feb 23rd Minhyong Kim (UCL)

March 9th Jean Gillibert (Bordeaux) "On Tate-Shafarevich groups and class groups"

Abstract: We discuss here a link between Shafarevich-Tate groups of certain elliptic curves and class groups of the fields on which these are defined. The curves we consider are endowed with a rational cyclic isogeny of prime degree. Basically, the connection is established via the so-called class group pairing (defined by Mazur and Tate). We note that similar constructions have already been used by various authors when performing descent, but with a different approach. March 16th Christopher Daw (UCL)

March 23rd Alexei Skorobogatov (Imperial)

## Autumn 2010

This term, the seminar was held at Imperial College and was organised by Ambrus Pal.

#### The seminar program for Autumn 2010

 6 October Ambrus Pal (Imperial)Crystalline Chebotarev density theorems 13 October Ivan Fesenko (Nottingham) Two adelic structures on arithmetic surfaces and the Tate version of the BSD conjecture. 20 October Lenny Taelman (Leiden) 27 October Daniel Caro (Caen) 3 November Lorenzo Ramero (Lille) Cohomological epsilon factors and p-adic analytic geometry. 8 November The London-Paris Number Theory Seminar speakers: F.Loeser (ENS, Paris), J.Pila (Bristol), B.Zilber (Oxford). 10 November Jean-Louis Colliot-Thelene (Orsay) Unramified cohomology in degree 3. 17 November Pierre Lochak (Jussieu) Topological methods in Grothendieck-Teichmueller theory. 24 November Jon Pridham (Cambridge) 1 December Tomer Schlank (Jerusalem) 8 December Tony Scholl (Cambridge)Hypersurfaces and the Weil conjectures 15 December TBA

## Summer 2010

This term, the seminar was held at Kings College and was organised by Payman Kassaei and Manuel Breuning.

#### The seminar program for Summer 2010

 12 May Matthias Flach (Caltech)"Weil-etale cohomology of regular arithmetic schemes" 19 May Jared Weinstein (UCLA)"Resolution of singularities on the Lubin-Tate tower" Abstract: A fundamental result in local class field theory is the 1965 paper of Lubin and Tate, which classifies the abelian extensions of a nonarchimedean local field in terms of an algebraic structure known as a one-dimensional formal module. We'll review this result, and show how the question of constructing nonabelian extensions leads to the study of the Lubin-Tate tower, which can be viewed as an infinitesimal version of the classical tower of modular curves $X(p^n)$. By results of Harris-Taylor and Boyer, the cohomology of the Lubin-Tate tower encodes precise information about non-abelian extensions of the local field (namely, it realizes the local Langlands correspondence). The Lubin-Tate tower has a horribly singular special fiber, which hinders any direct study of its cohomology, but we will show that after blowing up a singularity there is a model for the tower whose reduction contains a very curious nonsingular hypersurface defined over a finite field -- curious because it seems to have the maximum number of rational points relative to its topology. We will write down the equation for this hypersurface and formulate a conjecture (alas, still unproved) regarding its zeta function. 26 May Antonio Lei (Cambridge)"Wach modules and Iwasawa theory for modular forms II" Abstract: This is a follow-up of Sarah Zerbes' talk from last term. In Sarah's talk, she talked about the construction of Coleman maps using the theory Wach modules in order to reformulate Kato's main conjecture for modular forms at supersingular primes under some technical conditions. In this talk, I will introduce a new result on elementary divisors for Wach modules and Dieudonne modules which enables us to remove many of the assumptions in our previous works. 2 June The London-Paris Number Theory Seminar speakers: F. Brown, M. Hindry, M. Kakde 9 June Peter Swinnerton-Dyer (Cambridge)"Density of rational points on certain K3 surfaces" 16 June Shu Sasaki (King's College London)"On Artin representations and nearly ordinary Hecke algebras over totally real fields" Abstract: I will explain how to prove an analogue in the "completely split" Hilbert case of a result of Buzzard and Taylor about two-dimensional Artin representations and weight one forms, and prove new cases of the strong Artin conjecture for totally odd, two-dimensional "icosahedral" representations of the absolute Galois group of a totally real field. 23 June Kazim Buyukboduk (Istanbul)"Euler systems of rank $r$ and Kolyvagin systems" Abstract: For a $p$-adic Galois representation $T$, I will devise an Euler system / Kolyvagin system machinery which as an input takes an Euler system of rank $r$ (in the sense of Perrin-Riou), and gives a bound on the Bloch-Kato Selmer group in terms of an r x r determinant. I will give two fundamental applications of this refinement: The first with the (conjectural) Rubin-Stark elements; and the second with Perrin-Riou's (conjectural) $p$-adic $L$-functions.

## Spring 2010

This term, the seminar was held at University College and was organised by Andrei Yafaev.

#### The seminar program for Spring 2010

 13 January Michael Schein"On families of irreducible supersingular mod p representations of GL_2(F)" 20 January Mathieu Florence (Paris)"Equivariant birational geometry of Grassmannians" Abstract: Let k be a field, and A a finite-dimensional k-algebra. Let d be an integer. Denote by Gr(d,A) the Grassmannian of d-subspaces of A (viewed as a k-vector space), and by GL_1(A) the algebraic k-group whose points are invertible elements of A. The group GL_1(A) acts naturally on Gr(d,A) (by the formula g.E=gE). My aim is to study some birational properties of this action. More precisely, let r be the gcd of d and dim(A). Under some hypothesis on A (satisfied if A/k is etale), I will show that the variety Gr(d,A) is birationally and GL_1(A)-equivariantly isomorphic to the product of Gr(r,A) by an affine space (on which GL_1(A) acts trivially). By twisting, this result has a few corollaries in the theory of central simple algebras. For instance, let B and C be two central simple algebras over k, of coprime degrees. Then the Severi-Brauer variety SB(B \otimes C) is birational to the product of SB(B) \times SB(C) by an affine space of the correct dimension. These corollaries are in the spirit of Krashen's generalized version of Amitsur's conjecture. 27 January Fernando Villegas (Texas)"Hypergeometric motives" 3 February Toby Gee (Harvard)"The Sato-Tate conjecture for Hilbert modular forms" Abstract: I will discuss the Sato-Tate conjecture for Hilbert modular forms, which I recently proved in collaboration with Thomas Barnet-Lamb and David Geraghty. 10 February Jonathan Pila (Bristol)"A model-theoretic approach to problems of Manin-Mumford-Andre-Oort-type" Abstract: I will describe a result, joint with Alex Wilkie, about the distribution of rational points on certain non-algebraic sets in real space. The natural setting is an 'o-minimal structure over the real numbers', a notion from model-theory. A surprising strategy, proposed by Umberto Zannier, uses this result to approach diophantine problems in the Manin-Mumford-Andre-Oort circle of conjectures. I will describe some implementations of this strategy, including an unconditional proof of the Andre-Oort conjecture for products of modular curves. 17 February Frank Neumann (Leicester)"Moduli stacks of vector bundles on curves and Frobenius morphisms" Abstract: After giving a brief introduction into moduli problems and moduli stacks I will indicate how to calculate the l-adic cohomology ring of the moduli stack of vector bundles on an algebraic curve in positive characteristic and explicitly describe the actions of the various geometric and arithmetic Frobenius morphisms on the cohomology ring. It turns out that using the language of algebraic stacks instead of geometric invariant theory this becomes surprisingly easy. If time permits I will indicate how to prove some analogues of the classical Weil conjectures for the moduli stack. This is work in progress with Ulrich Stuhler (Goettingen). 24 February Wansu Kim (Imperial)"Galois deformation theory for norm fields" 3 March Lawrence Breen (Paris)"Non-abelian and partly abelian cohomology theories" 10 March Sarah Zerbes (Exeter)"Wach modules and Iwasawa theory for modular forms" 17 March Cecile Armana (Paris/Barcelona)"Coefficients of Drinfeld modular forms and Hecke operators" Abstract: We will talk about Drinfeld modular forms, which are analogues, over the function field $\mathbf{F}_{q}(T)$, of classical modular forms. Given a classical cusp form $f$, there exists a simple formula relating the $n$-th Fourier coefficient of $f$ to the first coefficient of $T_{n}(f)$ ($T_n$ denotes the $n$-th Hecke operator). This property has several consequences, for instance the multiplicity one theorem. Drinfeld modular forms possess series expansion and Hecke operators acting on them. The aim of the talk is to present a formula giving, for any Hecke Drinfeld eigenform, some of its coefficients in terms of its eigenvalues.

## Autumn 2009

This term, the seminar was held at Imperial College and was organised by Alexei Skorobogatov.

#### The seminar program for Autumn 2009

 7 October Samir Siksek (Warwick)"Explicit Chabauty over Number Fields" Abstract: Let $C$ be a curve of genus at least $2$ over a number field $K$ of degree $d$. Let $J$ be the Jacobian of $C$ and $r$ the rank of the Mordell-Weil group $J(K)$. Chabauty is a practical method for explicitly computing $C(K)$ provided $r \leq g-1$. In unpublished work, Wetherell suggested that Chabauty's method should still be applicable provided the weaker bound $r \leq d(g-1)$ is satisfied. We give details of this and use it to solve the Diophantine equation $x^2+y^3=z^{10}$ by reducing the problem to determining the $K$-rational points on several genus $2$ curves over $K=\Q(\sqrt[3]{2})$. 14 October Florian Pop (University of Pennsylvania and the Newton Institute) "On the Ihara/Oda-Matsumoto Conjecture" Abstract: In his "Esquisse d'un programme", Grothendieck suggested that one should be able to give a non-tautological description of the absolute Galois group of the rationals via its action on the geometric fundamental group of "interesting" varieties. Similar was suggested/asked by Ihara, and a precise conjecture was made by Oda-Matsumoto. In my talk I plan to report on the status of the art of this problem. 21 October Imperial Commemoration day (no seminar) 28 October Lassina Dembele (Warwick) "Nonsolvable Galois number fields ramified at 2, 3 and 5 only" Abstract: In the mid 90s, Dick Gross proposed the following conjecture. Conjecture: For every prime p, there is a nonsolvable Galois number field K ramified at p only. For p>=11, this conjecture is a consequence of results of Serre and Deligne (using classical modular forms). In this talk, we will show that the conjecture is true for p=2, 3 and 5. The extensions K we constructed in those cases are obtained by using Galois representations attached to Hilbert modular forms. We will also outline a strategy to tackle the case p=7 using automorphic forms on U(3). 4 November Roger Heath-Brown (Oxford)"Counting points on cubic curves" Abstract: Given a smooth plane cubic curve C defined over the rationals, we are interested in upper bounds for the number of rational points of height at most B, say, which are uniform in the curve C. Two previous approaches will be described, along with a new hybrid version. 11 November Don Blasius (UCLA)"Asymptotic Fullness of Automorphic Galois Representations" Abstract: On a reductive group G over a number field, limit multiplicity theorems give the growth rate, as a function of suitably growing level, for the number of cusp forms $\pi$ which have given discrete series type at infinity. In this talk we look at some finer structure arising from the existence of Galois representations attached to such forms. Specifically, we ask whether the subset of those with largest Zariski closure has density one among all the forms. For some simple cases we prove the conjecture, or provide a positive density result. One proof of the latter uses a result about the asymptotic distribution of Hecke eigenvalues at a fixed unramified finite place, namely that this distribution is Plancherel measure. 16 November The London-Paris Number Theory Seminar speakers: M. Emerton, A. Skorobogatov, S. David 18 November Herbert Gangl (Durham)"Double zeta values and periods of modular forms" Abstract: We give new relations among double zeta values \zeta(r,s)=\sum_{m>n>0} m^{-r} n^{-s} and show that the structure of the Q-vector space of all relations among double zeta values of weight k is connected in several different ways with the structure of the space of modular forms of weight k on the full modular group. (Joint work with M.Kaneko and D.Zagier.) 25 November Fabien Trihan (Nottingham)"On the p-parity conjecture in the function field case" Abstract: Let F be a function field in one variable with field of constants a finite field of characteristic p>0. Let E/F be an elliptic curve over F. We show that the order of the Hasse-Weil L-function of E/F at s=1 and the corank of the p-Selmer group of E/F have the same parity (joint work with C. Wuthrich). 2 December Behrang Noohi (King's)"Galois cohomology of crossed-modules and cohomology of reductive groups" Abstract: A 2-group (or a crossed-module) is a categorified version of a group. Line bundles over a scheme, for instance, form the Picard 2-group. Galois cohomology of 2-groups can be used to give information about Galois cohomology of ordinary groups (via, say, certain long exact sequences). We discuss the basics of the theory and give some simple examples involving Picard and Brauer groups. We then explain Borovoi's application of these ideas to the study of Galois cohomology of reductive groups. 9 December Javier Lopez (Queen Mary)"Torified schemes and geometry over the field with one element" Abstract: In this talk we introduce the notion of torified variety as a reduced scheme X of finite type that admits a decomposition $T = \{T_i\}_{i\in I}$ by split tori. This is a general concept that includes toric varieties, homogeneous spaces and Chevalley group schemes among others. We will show some of the main properties of torified varieties, show how the torifications define geometries over the field with one element. We also show how a torification provides an easy way to compute the counting function of $X$, which can be immediately applied to compute the corresponding zeta functions over $\mathbb{F}_1$.

## Summer 2009

This term, the seminar was held at King's College and was organised by Fred Diamond.

#### The seminar program for Summer 2009

 8 April Mehmet Haluk Sengun (Duisburg-Essen)"Computing With Bianchi Modular Forms" Abstract: Bianchi modular forms are modular forms over imaginary quadratic fields. In this talk, we present an algorithm to compute these forms and the Hecke action on them. Then we discuss their conjectural connections with mod p Galois representations, presenting certain results and calculations. 29 April Francis Brown (Paris-Jussieu)"Feynman graphs, moduli spaces and multiple zeta values" Abstract: I will begin by explaining how Feynman graphs in perturbative quantum field theory define interesting periods in the sense of algebraic geometry. Extensive computations by physicists suggest that these evaluate numerically to multiple zeta values in all known cases, but recent work of Belkale and Brosnan leads one to expect that the underlying motives may be of general type. After giving an overview of recent work on the subject, I will try to give a geometric and combinatorial explanation for these observations. 6 May Alexander Stasinski (Southampton)"Unramified and Regular Representations" Abstract: The talk will be about two rather different constructions of smooth (complex) representations of certain compact $p$-adic groups. The first is a cohomological construction of so called unramified representations of reductive groups over finite local rings, and is a generalization of the classical construction of Deligne and Lusztig. This gives in particular a family of representations of any compact $p$-adic group of the form $G(\mathfrak{o})$, where $G$ is a reductive group over the ring of integers $\mathfrak{o}$ in a local non-Archimedean field. The second construction is a purely algebraic approach to the regular representations of $GL_N(\mathfrak{o})$, which is formally similar to the Bushnell-Kutzko construction of supercuspidal representations. We shall describe the main features of the constructions, and discuss some open questions regarding their overlap, that is, to what extent representations given by one construction are also given by the other. 13 May Alex Bartel (Cambridge)"On class number relations in dihedral extensions of number fields" Abstract: In 1950, Brauer and Kuroda independently considered relations of class numbers and of regulators of intermediate fields in Galois extensions. These relations arise from the analytic class number formula and Artin formalism for L-functions and allow one to express certain quotients of class numbers in terms of corresponding quotients of regulators and of numbers of roots of unity. In some special cases, the regulator quotient can then be interpreted as a unit index. For extensions with dihedral Galois group of order 2p for p an odd prime, this was first done by Halter-Koch over Q and more recently by Lemmermeyer over arbitrary fields but under a very restrictive assumption. I will show how to derive a formula for arbitrary dihedral extensions of order 2p^n. The technique, which is purely representation theoretic, comes from the theory of elliptic curves, where one can consider a similar compatibility of the Birch and Swinnerton-Dyer conjecture with Artin formalism. 20 May Andreas Langer (Exeter)"Torsion zero cycles and p-adic integration theory" Abstract: We study the Chow-group of zero-cycles on the self-product of a CM-elliptic curve over the field of p-adic numbers and prove that its p-primary torsion subgroup is finite, provided that p is an ordinary good reduction prime and the p-adic L-function L_p(E,s) does not vanish at s=0. In the course of the proof we construct a new indecomposable element in K_1 which is integral at p, by using Coleman's p-adic integration theory and Besser's computation of syntomic regulators for K_2 of curves and K_1 of surfaces. 27 Maytwo talksroom 2B08 2:00-3:00, room 2B08 Nigel Boston (UC-Dublin, Wisconsin) "The fewest primes ramifying in a G-extension of Q" Abstract: If G is a finite group, what is the smallest number of primes ramifying in a G-extension of the rationals? We give evidence for a conjectural answer, together with a conjectural density for such n-tuples. [Parts are joint work with Ellenberg-Venkatesh and Markin.] 3:30-4:30, room 2B08 Jean-Pierre Serre"Variation with p of the number of solutions mod p of a given family of equations" 3 June  - 4 June The London-Paris Number Theory Seminar at King's College London, room 2B08 theme: p-adic modular forms speakers: Buzzard, Fargues, Loeffler, Colmez, Mokrane, Dimitrov, Panchishkin 10 Junecancelled cancelled The previously announced talk by Lassina Dembele is cancelled because of delays in getting his visa. 17 June Christopher Deninger (Muenster)"Vector bundles on p-adic curves and p-adic representations" Abstract: The classical Narasimhan-Seshadri correspondence gives a bijection between stable vector bundles of degree zero on a compact Riemannian surface and irreducible unitary representations of its fundamental group. In joint work with Annette Werner we have transferred this correspondence to some extent to a p-adic setting. We will report on recent progress and the main open questions in this area. There is related work of Faltings on p-adic Higgs bundles.

## Spring 2009

This term, the seminar was held at University College and was organised by Andrei Yafaev.

 14 January Victor Abrashkin (Durham)"p-adic semistable representations and generalization of the Shafarevich Conjecture" Abstract: Breuil's theory of semistable p-adic representations is applied to prove the following property: if X is a projective variety over Q with semistable reduction modulo 3 and good reduction at all other primes then its Hodge number h^{2,0} = 0. 21 January no seminar (Minhyong Kim's inaugural lecture "On numbers and figures" in room 505, Mathematics Department, UCL at 4.30pm) 28 January Anna Cadoret (Bordeaux)"A uniform open image theorem for p-adic representations of etale fundamental groups of curves" 4 February Tim Browning (Bristol)"Rational points on cubic hypersurfaces" Abstract: Given a cubic hypersurface X defined over Q, the circle method furnishes a method for establishing the existence of Q-rational points on the hypersurface, provided that the dimension is sufficiently large. Thanks to work of Davenport, and more recently of Heath-Brown, we can now treat cubic hypersurfaces of dimension at least 12. In this talk I show how this can be improved to dimension 11 when the underlying cubic form can be written as the sum of two forms without any variables in common. 11 February Pierre Parent (Bordeaux) "Method of Runge and modular curves" 18 February Adrian Diaconu (Nottingham) "Trace formulas and moments of automorphic L-functions" 25 FebruaryTWO TALKS 2:00-3:00 Hershy Kisilevsky (Concordia) "Critical values of derivatives of (twisted) elliptic L-functions" Abstract: Let $L(E/\Q,s) be the$L$-function of an elliptic curve$E$defined over the rational field$\Q.$If$\chi$is a Dirichlet character of odd prime order such that$L(E,1,\chi)=0,$we examine the special values of the derivative. If$L'(E,1,\chi)$is non-zero, we provide computational evidence for an "explicit formula" for its value. We also have some cases of higher order special values in the case that$\ord_{s=1}L(E,s,\chi)>1.$4:00-5:00 Christian Wuthrich (Nottingham) "Self-points on Elliptic Curves" Abstract: Let$E$be an elliptic curve of conductor$N$. Given a cyclic subgroup$C$of order$N$in$E$, we construct a modular point$P_C$on$E$, called self-point, as the image of$(E,C)$on$X_0(N)$under the modular parametrisation$X_0(N)\to E$. In many cases (e.g. when E is semi-stable), one can prove that the point is of infinite order in the Mordell-Weil group of$E$over the field of definition of$C$. The study of these points in the$PGL_2(\mathbb{Z}_p)$-tower inside$\mathbb{Q}(E[p^\infty])$continues earlier work of Harris. It is also possible to construct derivatives'' \a la Kolyvagin. 4 MarchTWO TALKS 2:00-3:00 Pierre Debes (Lille) "Specializations of Galois covers" Abstract: The motivation is to investigate the specializations of a Galois cover over some field. The rational points on some twisted cover provide a key to the problem. Good behaviour of these twists with respect to reduction leads to some concrete answer over "big" fields. Our results relate to some questions in inverse Galois theory, to some works of Fried, Colliot-Thelene, Ekhedal on Hilbert's irreducibility theorem and to some classical theorems of Grunwald and Neukirch. (This is a joint work with Nour Ghazi). 4:00-5:00 Gunter Harder (Bonn) "Denominators of Eisenstein classes" 9 March(Monday) London Number Theory Seminar, Special Lectures Bao Chau Ngo (Institute for Advanced Study) Lecture 1: "Fundamental lemma and Hitchin fibration" at 3:00 pm, room 706 Lecture 2: "Symmetry of Hitchin fibration and endoscopy" at 4:30 pm, room 706 11 March Ambrus Pal (Imperial) "Rational points on genus one curves" 13 March(Friday) London Number Theory Seminar, Special Lectures Bao Chau Ngo (Institute for Advanced Study) Lecture 3: "Decomposition theorem in the case of the Hitchin fibration" at 4:00 pm, room 500 18 March Richard Hill (UCL) "Residually infinite extensions of arithmetic groups" 25 March Victor Snaith (Sheffield) "Computing the Borel regulator" in room 505 ## Autumn 2008 This term, the seminar was held at Imperial College and was organised by Kevin Buzzard. #### The seminar program for Autumn 2008  1 October14.00-15.00room 342 Thomas Zink (Bielefeld) "p-divisible groups over regular local rings of mixed characteristics" 8 October Go Yamashita"Upper bounds for the dimensions of p-adic multiple zeta values" 15 October Gihan Marasingha (Bristol)"A degree 4 del Pezzo surface: Manin's conjecture and almost primes" 22 October Cecilia Busuioc (Imperial)"Milnor K-theory and Modular Symbols" 29 October Tejaswi Navilarekallu (Vrije Universiteit Amsterdam) "Equivariant p-adic L-values" 5 November Owen Jones (Imperial) "Analytically induced representations and generalised Verma modules" 12 November Seidai Yasuda (Kyoto) "Diagonal periods of GL(n) over the rational function field" 17 November(Monday) The London-Paris Number Theory Seminar at the Insitut Henri Poincaré in Paris 19 November Jeanine van Order (Cambridge) "Analogues of Rohrlich's theorem" 26 November Kevin McGerty (Imperial) "A gentle introduction to the geometric Langlands program" 3 December Mohamed Saidi (Exeter) "On Grothendieck's anabelian section conjecture for curves" 10 December Gaetan Chenevier (ENS Paris) "The infinite fern of Galois representations of type U(3)" 17 December Payman Kassaei (King's) "Geometry of Hilbert modular varieties and canonical subgroups of abelian varieties with real multiplication" ## Summer 2008 This term, the seminar was held at King's College and was organised by David Solomon. #### The seminar program for Summer 2008  23 April Henri Johnston (Oxford)"Non-existence and splitting theorems for normal integral bases" Abstract: This is joint work with Cornelius Greither. We establish new conditions that prevent the existence of (weak) normal integral bases in tame Galois extensions of number fields. This leads to the following result: under appropriate technical hypotheses, the existence of a normal integral basis in the upper layer of an abelian tower Q \subset K \subset L forces the tower to be split in a very strong sense. 30 April Richard Hill (UCL and Heilbronn Institute)"Vanishing theorems for p-adic automorphic forms" 7 May - 8 May The London-Paris Number Theory Seminar 13 May Tuesday 3:30-4:30 room 436 Cristian Popescu (UCSD)"On the Coates-Sinnott Conjectures" 14 May Andrew Booker (Bristol)"Computing automorphic forms on GL(3)" Abstract: My student, Ce Bian, announced the computation of a few "generic" rank 3 automorphic forms (meaning they are not lifts from lower rank examples) at the AIM workshop "Computing arithmetic spectra" in March. I will give a brief introduction to the theme of the workshop and describe Bian's computations. I'll also say a few words about the bewildering amount of attention that the work received subsequently. 21 May Ambrus Pal (Imperial College)"The Manin constant of elliptic curves over function fields" Abstract: We study the p-adic valuation of the values of normalized Hecke eigenforms attached to non-isotrivial elliptic curves defined over function fields of transcendence degree one over finite fields of characteristic p. Under certain assumptions we derive lower and upper bounds on the smallest attained valuation in terms of the minimal discriminant. As a consequence we show that the former can be arbitrarily small. We also use our results to prove for the first time the analogue of the degree conjecture unconditionally for infinite families of strong Weil curves defined over rational function fields. 28 May Gautam Chinta (City College of New York)"Sums of two squares and sums of three squares" Abstract: I will begin by describing three results -- Gauss's three squares theorem, Hamburger's converse theorem, and Maass's evaluation of a sum over Heegner points of the Eisenstein series for the modular group. I then will describe conjectural generalizations of these results to GL(3). The unifying theme is a conjecture of Jacquet on orthogonal periods of automorphic forms on GL(r). This is a joint work with Omer Offen. 4 June Doug Ulmer (U. Arizona at Tucson, and Paris)"On Mordell-Weil groups of abelian varieties over function fields" Abstract: I will sketch a construction which, among other things, relates CM of certain abelian varieties over a field k to Mordell- Weil groups of certain abelian varieties over K=k(t). The construction yields completely explicit Mordell-Weil groups of arbitrarily large rank for finite k and a less explicit, but new, construction of abelian varieties over K of moderately large rank when k is the field of algebraic numbers. 11 June Steven Galbraith (Royal Holloway)"Applications of the Frobenius map in elliptic curve cryptography" Abstract: Elliptic curves over finite fields provide groups for which the discrete logarithm problem seems to be hard. Hence, elliptic curves have applications in public key cryptography. The Frobenius map has been used to speed up arithmetic on elliptic curves. The talk will survey some of these ideas. We will also discuss security implications of using Frobenius maps and present a new algorithm for solving the "Frobenius expansion discrete logarithm problem" 18 June Mahesh Kakde (Cambridge)"On the non-commutative Main Conjecture for totally real number fields" 25 June Jonathan Pila (Bristol)"Rational points of definable sets and the Manin-Mumford conjecture" Abstract: I will discuss problems and results concerning the distribution of rational points on certain non-algebraic sets. More specifically, definable sets in o-minimal structures. I will describe a result, joint with Wilkie, that such a set X can have only few'' rational points in a suitable sense, that do not lie on some connected semi-algebraic subset of X of positive dimension. I will describe some further results and conjectures, connections with transcendence theory, and a new proof (with Zannier) of the Manin-Mumford conjecture by combining these ideas with a result of Masser. ## Spring 2008 This term, the seminar was held at University College and was organised by Minhyong Kim. #### The seminar program for Spring 2008  9 January Pierre Debes (Lille)"Inverse Galois theory, Abelian varieties and modular towers" Abstract: This is a joint work with Anna Cadoret. We show a new constraint in constructing Galois covers of$\P^1$over$\Q$with a given Galois group$G$. If for some prime$p$, the order of the abelianization$P^{ab}$of the$p$-Sylow subgroups$P$of$G$is suitably large, compared to the index$[G:P]$and the number$r$of branch points, then the branch points must coalesce modulo small primes. This is related to some rationality questions on the torsion of abelian varieties. This connection also provides a new viewpoint and new results on the Modular Tower program. 16 January Makis Dousmanis (Paris 13)"Reductions of some families of two-dimensional crystalline representations" 23 January Konstantin Ardakov (Nottingham)"Reflexive ideals in Iwasawa algebras" 30 January Ben Green (Cambridge)"Distribution of Polynomials over finite fields" Abstract: Let F be a finite field and consider polynomials P : F^n -> F in n variables. What can we say about polynomials which are not equi-distributed, i.e. for which the proportion of x for which P(x) = c is not roughly 1/|F|? We introduce a notion of "rank" for multi-variable polynomials and show that such polynomials must have low rank. We apply this result to the study of so-called Gowers norms. Let P : F^n -> F be any function. It is well-known that P is a polynomial of degree d if the polynomial Q obtained by differencing d+1 times is identically zero. What if Q is not identically zero, but merely biased towards zero? The inverse conjecture for the Gowers norms predicted that, in this case, P correlates with a degree d polynomial. Using the result described above we establish this in certain cases. We will also discuss an example which shows that the conjecture can fail in very low characteristic. Joint work with T. Tao. 6 February two talks4 pm-6 pm 4 pm-5 pm: Takako Fukaya (Keio and Cambridge)"Root numbers, Selmer groups, and non-commutative Iwasawa theory" 5 pm-6 pm: Kazuya Kato (Kyoto and Cambridge)"Classifying spaces of mixed Hodge (resp. p-adic Hodge) structures" 13 February Nick Shepherd-Barron (Cambridge)"Thomae's formula for non-hyperelliptic curves" Abstract: In 1857 Thomae gave formulae for the theta-constants of a hyperelliptic curve in terms of projective data of the curve. In this talk we explain what this means in terms of moduli spaces and extend it to non-hyperelliptic curves. 20 February Fabien Trihan (Nottingham)"Crystalline representations and F-crystals" 27 February Andreas Doering (Imperial)"Topos theory in the foundations of physics" 5 March Burt Totaro (Cambridge)"Moving codimension-one subvarieties over finite fields" 12 March Tamas Hausel (Oxford)"Arithmetic harmonic analysis on character and quiver varieties" ## Autumn 2007 This term, the seminar was held at Imperial College and was organised by Ambrus Pal. #### The seminar program for Autumn 2007  3 October David Loeffler (ICL)"Overconvergent p-adic automorphic forms and eigenvarieties for compact reductive groups" Abstract: I shall describe a construction of an eigenvariety parametrising p-adic automorphic forms for any reductive group G over Q that is split at p and compact at infinity. The construction generalises the work of Chenevier for compact forms of GL_n and Buzzard for quaternion algebras. The method gives a space of automorphic forms for each standard parabolic subgroup P of G; in this gives a hierarchy of "semi-classical" automorphic forms intermediate between the space of classical forms (corresponding to P = G) and the spaces constructed by Chenevier in the unitary case (which correspond to P = Borel). If there is time, I shall also mention ongoing work on classicality criteria and connections to Galois representations. 10 October Yiannis Petridis (UCL)"On the distribution of modular symbols" 17 October Andreas Schweizer (University of Exeter)"On the torsion of elliptic curves over sufficiently general function fields" Abstract: If K varies over all complex function fields and E varies over all elliptic curves over K with j(E) not in C, it is known that the size of the torsion group E(K)_{tors} can be uniformly bounded by a number depending only on the genus of K. Moreover, if one restricts to function fields K that are special'', for example hyperelliptic, one can even give absolute bounds (not depending on the genus of K) for the size of E(K)_{tors}. We will discuss what happens if K varies over all function fields that are sufficiently general''. 24 October David Solomon (King's College)"Stickelberger's Theorem Revisited" Abstract: Stickelberger's Theorem (from 1890) gives an explicit ideal in the Galois group-ring which annihilates the imaginary part of the class group of an abelian field. In the 1980s Tate and Brumer proposed a generalisation (the "Brumer-Stark conjecture" ) for in any abelian extension of number fields K/k with K CM and k totally real. Both the theorem and the conjecture leave certain questions unanswered: Is the (generalised) Stickelberger ideal the full annihilator, the Fitting ideal or what? And, at a more basic level, what can we say in the plus part, eg for a real abelian field? (In the latter case, Stickelberger's theorem amounts to little more than 0=0!) I shall discuss possible answers, some still conjectural, to pieces of these puzzles, using two new p-adic ideals of the group ring. There are interesting connections with Iwasawa Theory, the Equivariant Tamagawa Number Conjecture etc. 31 October Samir Siksek (University of Warwick)"Chabauty for Symmetric Powers of Curves" Abstract: Chabauty is a classical method for computing the rational points of curves of higher genus. In this talk, we explain an adaptation of Chabauty which allows us in many cases to compute all rational points on the d-th symmetric power of a curve provided the rank of the Mordell-Weil group of the Jacobian is at most g-d (where g is the genus). We illustrate this by giving two examples of genus 3, one hyperelliptic and the other plane quartic. 7 November Urs Hartl (University of Münster)"Period Spaces for Hodge-Structures in Equal Characteristic" Abstract: We construct period spaces for Hodge structures in equal characteristic. These Hodge structures were invented by Pink. The period spaces are analogues of the Rapoport-Zink period spaces for Fontaine's filtered isocrystals in mixed characteristic. For our period spaces we determine the image of the period morphism as a Berkovich open subspace. We prove the analogue of a conjecture of Rapoport Zink stating the existence of interesting local systems on this image. Moreover, we prove the analogue of the Colmez-Fontaine Theorem that "weakly admissible implies admissible". As a consequence the Berkovich open subspace mentioned above contains every classical rigid analytic point of the period space. 12 November The London-Paris Number Theory Seminar 14 November Alberto Minguez (University of East Anglia)"On the Howe correspondence" Abstract: The aim of this talk is to introduce the audience to the theory of local Howe correspondence. For the dual pair of type (Gl(n), Gl(m)) we will show a new proof which allows us to describe the correspondence in terms of Langlands parameters. At the end, we will discus about the possibility of having such a correspondence for l-modular representations. 21 November 4 pm-5:30 pm Eyal Goren (McGill University, Montréal) "Class invariants for CM fields of degree 4" Abstract: The problem of effective construction of units in abelian extensions of number fields is at present out of reach. Notwithstanding conjectural constructions, the only exceptions are the constructions for abelian extensions of Q and of a quadratic imaginary field, where the units are the cyclotomic and elliptic units respectively. One reason one seeks such constructions is to find Stark units which appear in Stark's conjectures on special values of L functions. In this talk, after explaining what is the source of the difficulty, I shall survey what we know at present about the case of CM fields of degree 4, focusing on my work with Ehud de Shalit, Kristin Lauter and Daniel Vallieres. Time allowing, I shall try and put the results in the perspective of the work of Jan Bruinier and Tonghai Yang, indicate some proofs of our results and discuss work in progress. 28 November Carlos Castano-Bernard (ICTP, Trieste)"On the subgroup generated by the traces of Heegner points on elliptic curves" Abstract: Consider an elliptic curve E over Q and assume its L-function has a simple zero at s = 1. In particular, there is a non-constant morphism X0(N)/wN --> E defined over Q, where wN is the Fricke involution and N is the conductor of E. So the trace of each Heegner point on E is Q-rational. Moreover, it is well-known that E(Q)/E(Q)tors is isomorphic to Z, and in fact the images of the traces in E(Q)/E(Q)tors generate a subgroup of finite index I. In this talk we shall discuss a conjecture that predicts that whenever N is prime and the index I > 1, then the real locus (X0(N)/wN)(R) has more than one connected component or--less likely--the Tate-Shafarevich group of E is non-trivial. 5 December Laurent Fargues (Université Paris-Sud, Orsay)"Ramification of Lubin-Tate groups and the Bruhat-Tits building" Abstract: One of the purposes of this talk is to give a description of the isomorphism between the p-adic Lubin-Tate and Drinfeld towers at the level of their skeletons. For the Drinfeld space, its skeleton is the Bruhat-Tits building of the linear group. For example, we can describe explicitly the pull-back of this simplicial structure on the open p-adic ball associated to the Lubin-Tate space. We also study in detail the different ramification filtrations (upper and lower) associated to Lubin-Tate groups. We give applications to generalized canonical subgroups and fundamental domains for Hecke correspondences. 12 December Sarah Zerbes (ICL)"Formulae for the higher Hilbert pairing" ## Summer 2007 This term, the seminar was held at University College and was organised by Richard Hill. #### The seminar program for Summer 2007  25 April Shaun Stevens (UEA)"Supercuspidal representations of p-adic classical groups" 2 May The London-Paris Number Theory Seminar 11 am - 4:30 pm, Imperial College London 9 May Ben Smith (Royal Holloway)"Computing Explicit Isogenies" Abstract: Isogenies --- surjective homomorphisms of algebraic groups with finite kernel --- are basic objects in number theory. Algorithms for computing with isogenies of elliptic curves are well-known; in higher dimensions, however, the situation is more complicated, and few explicit non-trivial examples of isogenies are known. We will describe some interesting examples of explicit isogenies of Jacobians of low-genus curves, discuss some of the computational issues, and give some applications in modern cryptography. 16 May Matthias Strauch"Potentially crystalline representations and associated p-adic representations of GL_2" Abstract: This is about joint work in progress with C. Breuil. For a certain family of potentially crystalline but not semistable two-dimensional representations of the absolute Galois group of Q_p we construct locally analytic representations of GL_2(Q_p), naturally parameterised by the Galois representations. 23 May Manuel Breuning (KCL)"Determinant functors and Euler characteristics" 30 May Jayanta Manoharmayum (Sheffield)"Lifting Galois representations" ## Spring 2007 This term, the seminar was held at King's College and was organised by David Burns. #### The seminar program for Spring 2007  17 January Kevin Buzzard (Imperial)"Mod p Galois representations and modular forms" 24 January Richard Hill (UCL)"Singular cohomology of modular curves" Abstract: Let \Gamma be an arithmetic group acting on the upper half-plane H, either with cusps or cocompact. Let \Gamma' be a normal subgroup of \Gamma. Then the quotient group G=\Gamma/\Gamma' acts on the cohomology of \Gamma'. I'll describe the structure of H^1(\Gamma',\Z) as a \ZG-module. 31 January David Burns (King's)"Iwasawa theory of elliptic curves in p-adic Lie extensions" 7 February Alex Paulin (Imperial)"Local to Global Compatibility on the Eigencurve" Abstract: To any classical cuspidal eigenform one can attach both a smooth irreducible representation of GL_2(Q_l) and a two dimensional Frob-semisimple Weil-Deligne representation. Classical Local-Global compatibility ensures that these agree under the (correctly normalised) local langlands correspondence. I will discuss ways of attaching such objects to overconvergent p-adic eigenforms across the eigencurve and to what extent local-global compatibility remains valid. 14 February Tom Fisher (Cambridge)"Finding rational points on elliptic curves using 6-descent and 12-descent" Abstract: Descent on an elliptic curve E is used to obtain partial information about both the group of rational points (the Mordell-Weil group) and the failure of the Hasse principle for certain twists of E (the Tate-Shafarevich group). The Selmer group elements computed may be represented as n-coverings of E. Traditionally one takes n to be a prime power. Breaking with this tradition, I explain how to combine the data of an m-covering and an n-covering, for m and n coprime, to obtain an mn-covering. This technique improves the search for rational points on E. In particular using 6-descent and 12-descent, I was recently able to find all the "missing" generators for the elliptic curves of analytic rank 2 in the Stein-Watkins database. 21 February Daniel Delbourgo (Nottingham)"Non-abelian congruences between L-values of elliptic curves" 28 February Olivier Brinon (Paris 13)"Overconvergence of$p$-adic representations: the relative case (joint work with F. Andreatta)" 7 March Jan Kohlhaase (Muenster)"The Cohomology of locally analytic Representations" Abstract: Starting with smooth representations of a p-adic reductive group, we will recall what is meant by a supercuspidal representation and the role such representations play in the local Langlands correspondence. We will then pass to locally analytic representations in the sense of Schneider/Teitelbaum, sketch the construction of locally analytic cohomology and generalize the above notion of supercuspidality to locally analytic representations. In the end, we will compute the (higher) Jacquet modules of locally analytic principal series representations and indicate why this is of importance for the p-adic Langlands correspondence. 14 March Colin Bushnell (King's)"Characters and constants" 21 March Tobias Berger (Cambridge)"Congruences between modular forms over imaginary quadratic fields" Abstract: We present two applications of congruences involving Harder's Eisenstein cohomology classes. We first prove a lower bound for the size of the Selmer group of certain Galois characters of imaginary quadratic fields coinciding with the value given by the Bloch-Kato conjecture. We further show how to obtain instances of the Fontaine-Mazur conjecture for imaginary quadratic fields in the residually reducible case. The latter is joint work in progress with Kris Klosin. ## Autumn 2006 This term, the seminar was held at Imperial and was organised by Alexei Skorobogatov. #### The seminar program for Autumn 2006  4 October Andrei Yafaev (UCL)"On the triviality of rational points on certain Atkin-Lehner quotients of Shimura curves" Abstract: This is a joint work with Pierre Parent. We use a modification of Mazur's method to prove that the only possible rational points on certain Atkin-Lehner quotients of Shimura curves come from CM points. 11 October Alexei Skorobogatov (Imperial College)"A finiteness theorem for the Brauer group of K3 surfaces" Abstract: Let k be a field finitely generated over the rationals, and let X be a K3 surface over k. We prove that Br(X)/Br(k) is finite. 18 October Tim Dokchitser (Cambridge)"Parity of ranks for elliptic curves with a cyclic isogeny" Abstract: This is a joint work with Vladimir Dokchitser. Let E be an elliptic curve over a number field K which admits a cyclic p-isogeny and semistable at primes above p. Then one can determine the root number and the parity of the p-Selmer rank for E/K, in particular confirming the parity conjecture for such curves (with an extra mild assumption for p=2). 25 October no seminar (Imperial's Commemoration Day) 1 November Jan Nekovar (Paris 7)"Parity of ranks of Selmer groups in p-adic families" 8 November Sarah Zerbes (Imperial)"Higher-dimensional logarithmic derivatives" Abstract: In my talk, I will explain how to construct logarithmic derivative maps for n-dimensional local fields of mixed characteristic (0,p). The main ingredients for this construction are higher-dimensional rings of overconvergent series and Tony Scholl's work on general fields of norms. As an application of the logarithmic derivative, I will give a new construction of Kato's dual exponential map for K_n. 13 November Séminaire de théorie des nombres Londres-Paris à l'Institut Henri Poincaré programme Orateurs: Fred Diamond, Toby Gee, Florian Herzig. 15 November Andres Helfgott (Bristol) "How small must ill-distributed sets be?" (joint with A Venkatesh) Abstract: Consider a set$S\subset \mathbb{Z}^n$. Suppose that, for many primes$p$, the distribution of$S$in congruence classes$\mo p$is far from uniform. How sparse is$S$forced to be thereby? A clear dichotomy appears: it seems that$S$must either be very small or possess much algebraic structure. We show that, if$S\subset \mathbb{Z}^2 \cap \lbrack 0, N\rbrack^2$occupies few congruence classes$\mo p$for many$p$, then either$S$has fewer than$N^{\epsilon}$elements or most of$S$is contained in an algebraic curve of degree$O_{\epsilon}(1)$. Similar statements are conjectured for$S\subset \mathbb{Z}^n$,$n\neq 2$. We follow an approach that combines ideas from the larger sieve of Gallagher \cite{Ga} and from the work of Bombieri and Pila \cite{BP}. All techniques used are elementary. 22 November Fred Diamond (King's) "The weight part of Serre's conjecture for Hilbert modular forms" Abstract: I will explain the statement of a generalization of Serre's conjecture on mod p Galois representations to the context of Hilbert modular forms. The emphasis will be on the recipe for the set of possible weights (formulated by Buzzard, Jarvis and myself, and partly proved by Gee) and its behavior in some special cases. 29 November Ambrus Pal (Imperial)"On a conjecture about the cohomology of arithmetic groups" 6 December Jean Gillibert (Manchester) "Geometric Galois module structure and abelian varieties of higher dimension" Abstract: The so-called class-invariant homomorphism$\psi_n$, introduced by M. J. Taylor, measures the Galois module structure of (rings of integers of) extensions of the form$K(\frac{1}{n}P)/K$, where$K$is a number field,$P$is a$K$-rational point on an abelian variety$A$, and$n>1$is an integer. When$A$is an elliptic curve and$n$is coprime to 6, then$\psi_n$vanishes on torsion points. We explain here how, using Weil restrictions of elliptic curves, it is possible to construct abelian varieties of higher dimension for which this vanishing result is no longer true. 13 December Payman Kassaei (King's) "A subgroup-free" approach to Canonical Subgroups" Abstract: I will discuss joint work with E. Goren in which we present a subgroup-free'' approach to canonical subgroups, which in particular extends all aspects of the classical theory of canonical subgroups of elliptic curves to many various Shimura curves of interest. ## Summer 2006 This term, the seminar was held at KCL and was organised by David Burns. #### The seminar program for Summer 2006  3 May Alan Lauder (Oxford)"Zeta functions of surfaces over finite fields" 10 May Carlos Castano-Bernard (Cambridge)"Two approaches in the study of$X_0^+(N)(Q)$" Abstract: In this talk we discuss two approaches in the study of the rational points of$X_0^+(N)$. The first was suggested by certain geometric properties of rational points of the canonical embedding of$X_0^+(N)$in projective space, while the second approach was suggested by questions raised by Shimura on the connected components of the real locus of a given variety defined over the rational numbers. 17 May Ben Smith (Royal Holloway)"Examples of explicit isogenies" Abstract: In "Factorisations explicites de g(y) - h(z)", Cassou-Nogues and Couveignes give a classification of the pairs of irreducible polynomials g and h such that g(y) - h(z) is reducible. We describe their results, and show how they may be used to derive families of hyperelliptic Jacobians with an explicit isogeny. 24 May Rob de Jeu (Durham)"The Beilinson conjectures for$K_2$of curves" Abstract: The Birch-Swinnerton-Dyer conjecture relates the value of the L-function of an elliptic curve defined over a number field at 1 with its codimension 1 Chow group. The latter is part of$K_0$of the curve. Beilinson formulated conjectures of a similar flavour for all K-groups of any smooth projective variety defined over number fields, and we'll discuss those conjectures for$K_2$of curves. 31 May Herbert Gangl (Durham)"Double zeta values and modular forms" Abstract: We give new relations among double zeta values \zeta(r,s) = \sum_{m>n>0} m^{-r}n^{-s} and show that the structure of the Q-vector space of all relations among double zeta values of weight k is connected in several different ways with the structure of the space of modular forms of weight k on the full modular group. 7 June Guy Henniart (Paris)"Representations of p-adic GL(2), old and new" Abstract: Classical modular forms have their local counterpart, which are irreducible representations of GL(2,Q_p) on complex vector spaces. Congruences between modular forms lead to study continuous representations of GL(2,F), where F is a p-adic field, on non-Archimedean Banach spaces,or on fields of positive characteristic. I shall review recent work and conjectures, with special emphasis on the case of mod-p representations. 14 June Otmar Venjakob (Bonn)"On the leading terms of p-adic L-functions in non-commutative Iwasawa theory" Abstract: In the$GL_2$-main conjecture for elliptic curves without complex multiplication formulated by Coates, Fukaya, Kato, Sujatha and the speaker the existence of a (non-commutative)$p$-adic$L$-function$\mathcal{L}$is crucial satisfying an interpolation formula related to the value at$s=1$of certain twists of the Hasse-Weil$L$-function of$E$twisted by certain Artin representations. By the work of Fukaya and Kato the interpolation property and existence of$\mathcal{L}$is a consequence of their non-commutative Tamagawa number conjecture as well as a local analogue of the non-commutative Tamagawa number conjecture. The aim of this talk is to report on recent joint work with David Burns: We introduce leading terms of the$p$-adic$L$-function$\mathcal{L}$at Artin representations and extend Fukaya and Kato's interpolation formula now also involving the leading terms of the twisted complex Hasse-Weil$L$-functions. 21 June 2.30pm Minhyong Kim (Purdue)"Fundamental groups and Diophantine Geometry" 21 June 4.00pm Martin Taylor (Manchester)"Galois structure and Riemann-Roch theorems" Abstract: The talk will begin with a general discussion of the importance of Euler characteristics the study of arithmetic modules. We shall then consider the use of Riemann Roch theorems in the determination of Euler characteristics: firstly, in a fairly general situation of varieties over fields; then for various torsors over arithmetic varieties. 26 July 2.00pm Doug Ulmer (U. Arizona, Tucson)"Abelian varieties of large rank over function fields" Abstract: For every prime p and every integer g>0 we construct explicit abelian varieties of dimension g over F_p(t) for which the Birch and Swinnerton-Dyer conjecture holds and with arbitrarily large Mordell-Weil rank. ## Spring 2006 This term, the seminar was held at KCL and was organised by David Solomon. #### The seminar programme for Spring 2006  18 January Tim Dokchitser (Cambridge)"Ranks of elliptic curves in cubic extensions" Abstract: For an elliptic curve E over a number field K, I shall prove that the algebraic rank of E goes up in infinitely many extensions of K obtained by adjoining a cube root of an element of K. I shall also discuss how this relates to root numbers and Iwasawa theory, with E=X_1(11) over Q as a specific example. 25 January Andrei Yafaev (UCL)"The Andre-Oort conjecture" Abstract: This is a joint work with Bruno Klingler. We explain a proof of the Andre-Oort conjecture under the assumption of the generalised Riemann Hypothesis. 1 February David Burns (KCl)"Algebraic p-adic L-functions in non-commutative Iwasawa theory" Abstract: There have been several important developments in non-commutative Iwasawa theory over the last few years. We discuss a natural construction of algebraic p-adic L-functions in this setting and discuss some interesting consequences for the main conjectures of non-commutative Iwasawa theory formulated by Coates, Fukaya, Kato, Sujatha and Venjakob and by Fukaya and Kato. 8 February Daniel Caro (Durham)"Towards a good p-adic cohomology" Abstract: First, we will trace the history of the search for a good p-adic cohomology over schemes in characteristic p. We arrive at the construction of Berthelot's arithmetical D-modules. We will explain why these objects now represent the only possibility of obtaining a good p-adic cohomology. An important result which inspires trust in this theory is the following: for every overholonomic F-complex E of arithmetical D-modules over a variety X of characteristic p, there exists a splitting of X into locally closed subvarieties X _i such that the restrictions of E on X _i become much simpler (i.e., come from overconvergent F-isocrystals). In this talk, we will recall basic definitions and explain the meaning of this splitting. 15 February Ivan Horozov (Durham)"Euler characteristics of arithmetic groups" Abstract: The talk will be about Euler characterisitcs of the general linear group, the special linear group, and the symplectic group over rings of algebraic integers. I will present a method for computing the homological Euler characteristic, as well as some applications to values of Dedekind zeta function at -1 and at -3 and to Kummer and Greenberg criteria for divisibility of certain class numbers by a prime. 22 February Avner Ash (Boston College)"Symmetries of Algebraic Numbers" Abstract: The Absolute Galois Group of Q, notated G_Q acts on the solution sets of systems of polynomial equations with rational coefficients. Linear representations of G_Q play a leading role in getting information about the solution sets, e.g. in Wiles's proof of Fermat's Last Theorem. One influential conjecture in this area is that of Serre, concerning 2-dimensional representations of G_Q over finite fields. I will briefly review Serre's conjecture and discuss generalizations to higher dimensional representations. 1 March Emmanuel Kowalski (Bordeaux)"The algebraic principle of the large sieve" Abstract: Linnik's original large sieve gives upper bounds for the number of integers in an interval with reductions modulo primes restricted to fall in fairly small sets. The talk will describe an abstract sieve setting which can lead to such results in more general situations. Then applications to the average distribution of Frobenius elements in families of algebraic varieties over finite fields will be discussed, and some work in progress concerning arithmetic properties of integral unimodular matrices. In one case the Riemann Hypothesis of Deligne is the crucial ingredient, in the other the spectral theory of automorphic forms appears naturally. 8 March Misha Gavrilovich (Oxford)"Model theory, Z-extensions of C*, the exponential function, and a homotopy-theory viewpoint on some arithmetic issues" 15 March James McKee (Royal Holloway)"Salem numbers, Pisot numbers, graphs, and Mahler measure" Abstract: This is joint work with Chris Smyth.We use graphs to define sets of Salem and Pisot numbers, and prove that the union of these sets is closed, supporting a conjecture of Boyd that the set of all Salem and Pisot numbers is closed. We find all trees that define Salem numbers. We show that for all integers n the smallest known element of the n-th derived set of the set of Pisot numbers comes from a graph. We define the Mahler measure of a graph, and find all graphs of Mahler measure less than (1+sqrt5)/2. We start the task of extending this work from graph adjacency matrices to all integer symmetric matrices by classifying all such matrices having all eigenvalues in the interval [-2, 2]. 22 March Graham Everest (UEA) "Descent and Divisibility" Abstract: In 2001, Bilu, Hanrot and Voutier proved that for every n>30, the nth term of a Lucas or Lehmer sequence must have a primitive divisor. This is a remarkable result because of its uniform nature and the smallness of the bound 30. I will report on an elliptic analogue of their theorem. Also, I will report on the apparently harder statement about prime values. 5 April Werner Bley (University of Kassel) "Computation of class groups" 12 April Andrew Jones (KCL) "Dirichlet$L$-functions at$s=1$and Fitting invariants of ideal class groups" Abstract: We show that a special case of the equivariant Tamagawa number conjecture (ETNC) of Burns and Flach implies a refinement of a $p$-adic integrality conjecture' of Solomon concerning leading terms at$s = 0$of certain Twisted Zeta-functions'. In fact, the ETNC implies that an ideal defined by Solomon belongs to the initial Fitting invariant of a certain ideal class group. Since the relevant case of the ETNC is known to be valid for absolutely abelian fields we thereby obtain new results about the structure of ideal class groups. In particular, we obtain an analogue of Stickelberger's Theorem. ## Autumn 2005 This term, the seminar was held at Imperial and was organised by Toby Gee. #### The seminar programme for Autumn 2005  5 October Toby Gee (Imperial)"On the weights of mod p modular forms" 12 October Manuel Breuning (KCL)"A reinterpretation of the Chinburg conjectures" 19 October Adam Joyce (Imperial)"Stable models of modular curves" Abstract: Using the Drinfeldian notion of level structures, one can obtain integral moduli spaces over rings of cyclotomic integers, whose generic fibres are the well-known algebraic curves over Q, X_0(N) and X_1(N). These moduli spaces are not smooth in characteristics dividing the level. For applications to arithmetic geometry, it is advantageous to have stable models (in the sense of Deligne-Mumford) for algebraic curves. The moduli spaces above are often not stable models for their generic fibres. We describe certain stable models, using tools from algebraic geometry. 26 October David Solomon (KCL)"Stark Units, Hilbert Symbols and a Stickelberger Ideal at s=1" Abstract: Let M/k be an abelian extension of totally real number fields. For appropriately chosen finite sets of places S of k, Stark's conjecture predicts the existence of (an element of an exterior power of the) S -units of M whose equivariant regulator gives the leading term at s=0 of Artin L-functions L_S(s,\chi) for all characters \chi of G=Gal(M/k). These Stark Units' are cyclotomic units if k=Q but otherwise are not known to exist. If instead M is of CM type (and k is still totally real) the Brumer-Stark conjecture predicts that the values at s=0 of the L_S(s,\chi) now define an ideal of ZG that generalises the Stickelberger ideal (the case k=Q). In particular, it annihilates the odd' part of Cl(M). I shall discuss two p-adic conjectures that, in a sense, tie these two other conjectures together. For all odd \chi, the functional equation relates L_S(0,\chi) to L_S(1,\chi) . For each odd prime p, the latter values allow us to define a map the exterior power of the p-semilocal units of M into Q_pG . I conjecture firstly that the image of this map lies in Z_pG and secondly (when M contains p^nth roots of unity) that the map is congruent modulo p^n to one defined using Hilbert symbols and the (conjectural) Stark units coming from the maximal real subextension M^+/k. 2 November Cornelius Greither (Munich)"Fitting ideals of class groups via the Equivariant Tamagawa Number conjecture" 9 November Samir Siksek (Warwick)"Classical and modular approaches to exponential Diophantine problems" 16 November Denis Benois (Besan n)"Iwasawa theory of crystalline representations and$(\phi,\Gamma)$-modules" 23 November Tony Scholl (Cambridge)"Higher fields of norms and (phi,Gamma)-modules" 30 November 2.30pm Don Zagier (Bonn)"Double zeta values and modular forms" 30 November 4.00pm Richard Pinch "The distribution of Carmichael numbers" 7 December David Whitehouse (Caltech)"The twisted weighted fundamental lemma for the transfer of automorphic forms from GSp(4) to GL(4)" 14 December Tim Browning (Bristol)"Density of rational points on smooth hypersurfaces" Abstract: Let$X$be a non-singular projective hypersurface of degree$d>1$and dimension$k$. It has been conjectured that the number of rational points on$X$, which have height at most$B$, should be$O(B^{k+\eps})$for any$\varepsilon>0$. The implied constant here should be allowed to depend at most upon$d,k$and the choice of$\eps$. In this talk, which comprises joint work with Heath-Brown, we discuss the final resolution of this conjecture. ## Summer 2005 This term, the seminar was held at Imperial and was organised by Alexei Skorobogatov. #### The seminar programme for Summer 2005  20 April Toby Gee (Imperial)"New results for companion forms over totally real fields" 27 April Kazuhiro Fujiwara (Nagoya and Cambridge)"Galois representations over cyclotomic towers" 4 May Peter Swinnerton-Dyer (Cambridge)"Counting rational points and the Manin conjecture" 11 May Vassily Golyshev (Independent Moscow University)"Differential equations of quantum cohomology and higher Apery recurrences." Abstract: We say that a linear polynomial recurrence with integer coefficients is of Apery type if it has two solutions, the quotient of which b_n/a_n tends to an irrational zeta (or L-function) value at an integer point. We consider recurrences that are Mellin transforms of differential equations of quantum cohomology for Grassmannians. We conjecture that these recurrences are of Apery type. We prove certain cases of this conjecture for Grassmannians of classical groups by reducing the DEs in question to modular ones. 18 May Daniel Delbourgo (Nottingham)"Euler products over C_p" Abstract: Standard Euler products converge in some right half-plane Re(s)>constant. If one tries the same trick replacing the complex numbers with the p-adics, things quickly go wrong. We first explain a way of making Euler products converge over C_p, the Tate field. Fortunately, these products converge to the values of classical L-functions with appropriate modifications to the Euler factor at p. The proof uses fractional calculus and something resembling an explicit reciprocity law. If we've got time, we'll mention some convergence calculations for p-adic L-functions of elliptic curves. 25 May Chris Skinner (Michigan)"L-values, congruences, and Selmer groups" 1 June Alexei Skorobogatov (Imperial)"Global points on Shimura curves" Abstract: It is an open question whether all counterexamples to the Hasse principle on smooth projective curves over number fields are due to the Manin obstruction. In the 1980-s Bruce Jordan proved that global points don't exist on certain Shimura curves, producing counterexamples to the Hasse principle. I'll show how these and other known counterexamples are explained by the Manin obstruction. 8 June Andrei Yafaev (UCL)"Recent progress on the Andre-Oort conjecture" 15 June Teruyoshi Yoshida (Harvard and Nottingham)Compatibility of local and global Langlands correspondences" (with R.Taylor) Abstract: The work of Harris-Taylor, which proved the local Langlands correspondence for GLn, included the construction of l-adic Galois representations attached to certain class of automorphic representations of GLn over CM fields, compatible with the local Langlands correspondence up to semisimplification at all places outside l. By studying the semistable reduction of the relevant Shimura varieties, we strengthen this result to show that the local monodromies are also the correct ones. The irreducibility of global Galois representations follows. 22 June Tom Fisher (Cambridge)"Computing models for visible elements of the Tate-Shafarevich group" 29 June Bjorn Poonen (Berkeley and Cambridge)"Multiples of subvarieties in algebraic groups over finite fields" ## Spring 2005 This term, the seminar was held at KCL and was organised by David Solomon. #### The seminar programme for Spring 2005  12 January Michael Harris (Universite Paris VII)"Deformations of automorphic Galois representations" Abstract: The l-adic cohomology of Shimura varieties attached to certain unitary groups provide compatible systems of n-dimensional l-adic representations of the absolute Galois group of a CM field. These representations are necessarily polarized (self-dual, more or less) and their Hodge-Tate weights have multiplicity one. In joint work with Taylor we have proved, under the usual restrictions, that any representation with these properties arises in this way, provided (a) the reduction mod l has at least one modular lifting and (b) the representation is a minimal lifting of its reduction mod l. Work of R. Mann allows us to remove the minimality condition (b) under a precise conjecture on mod l modular forms known as Ihara's Lemma. 19 January Prof E.V. Flynn (Liverpool)"The Brauer-Manin Obstruction on Curves" Abstract: When a variety violates the Hasse principle, this can be due to an obstruction known as the Brauer-Manin obstruction. It is an unsolved problem whether all violations of the Hasse principle on curves are due to this obstruction. I shall describe work in progress which tests a wide selection of curves, and tries to decide for these examples whether the Brauer-Manin obstruction is the cause of all violations of the Hasse principle. This has involved the development of several new techniques, exploiting the embedding of a curve in its Jacobians via a rational divisor class of degree 1, and has produced examples of certain new types (in response to a request of Alexei Skorobogatov). If there is time, I shall also discuss the loosely related question of annihilation and visualisation of members of the Shafarevich-Tate group of Jacobians. 26 January Toby Gee (Imperial)"A new proof of an old theorem on companion forms" Abstract: Results on companion forms over Q were obtained in the early 1990s by Gross and Coleman-Voloch. I gave a generalisation to totally real fields last year. In this talk I will discuss a new and much more conceptual proof of the results of Gross and Coleman-Voloch, and indicate the possibilities for further generalisations to totally real fields. 2 February Prof Francis Johnson (UCL)"Orders in quaternion algebras and recent developments in algebraic homotopy theory" Abstract: It is a fundamental question in non-simply connected homotopy theory (and thereby also in combinatorial group theory) to decide whether, over a given fundamental group G, every algebraic 2-complex is geometrically realizable. Although for most finite G this problem is increasingly well understood, the quaternion groups Q_4n have proved to be exceptional. Using Swan's work on non-cancellation phenomena for modules over orders in quaternion algebras over number fields, we describe families of algebraic 2-complexes over Q_4n for which no geometric realisations are currently known (or, in terms of combinatorial group theory, which do not correspond to any known group presentation) 9 February Vladimir Dokchitser (Cambridge)"Root numbers and the rank of elliptic curves" Abstract: Fix an elliptic curve E over Q. I will discuss the behaviour of the sign in the functional equation of the L-function L(E/K,s), where K varies over different number fields. When the sign is -1, the L-function (assuming it is exists and is analytic) has a zero at s=1, and the Birch-Swinnerton-Dyer conjecture predicts that E should have a point of infinite order over the number field K. It is then possible to obtain examples of elliptic curves over Q which, while not having any rational points of infinite order, must conjecturally have points of infinite order over all the fields Q(m^{1/3}) for every (non-cube) m>1. I will discuss this, and similar phenomena. 16 February Andreas Langer (Exeter)"Gauss-Manin connection via Witt-differentials" Abstract: For a scheme X that is smooth over a p-adic base we show an equivalence of categories between the category of locally free crystals and the category of Witt-connections on X. The proof uses the relative de Rham-Witt complex and generalizes a recent result of Bloch. As an application we realize the Gauss-Manin connection in the de Rham-Witt complex. 23 February Kyu-Hwan Lee (U. Toronto)"Spherical Hecke algebras of GL_n over 2-dimensional local fields" Abstract: At the beginning of the talk, we will briefly review the classical Satake isomorphism, which plays an important role in the Langlands program. Then we will try to generalize the theory to the 2-dimensional local field case. More precisely, we will construct spherical Hecke algebras of GL_n over 2-dimensional local fields and prove the Satake somorphism for the algebras. We will use Fesenko's measure to define the Satake isomorphism. A connection to Kac-Moody groups will also briefly discussed. This is a joint work with Henry Kim. 2 March Jayanta Manoharmayum (Sheffield)"Minimal deformations of Galois representations" Abstract: I will describe how one can get, in some cases, lifts of residual representation which are minimally ramified. 9 March Ivan Fesenko (Nottingham)"Poles of the Hasse zeta function" Abstract: The talk will try to discuss some of the applications of the study of the Hasse zeta function of elliptic curves over global fields via 2d zeta integrals to: (a) Riemann hypothesis for the zeta function; (b) location of poles of the zeta function on the critical line; (c) an extension of the class of zeta functions all whose motivic L-factors are automorphic, using Laplace-Carleman transforms of odd mean periodic functions, and importance of this for the Langlands programme;(d) the rank part of the BSD conjecture. 16 March Jan Nekovar (Jussieu)"The Euler system of CM points" Abstract: We shall discuss a generalization of Kolyvagin's results on Heegner points. 23 March Steven Galbraith (Royal Holloway College)"Pairings on abelian varieties and cryptography" Abstract: The talk will survey some applications of elliptic curves over finite fields to cryptography. In particular, applications of the Weil and Tate pairings will be described and some new results on efficient computation of these pairings will be presented. ## Autumn 2004 This term the seminar was held at Imperial College and was organised by Toby Gee #### The seminar programme for Autumn 2004  27 October No seminar 3 November Pierre Parent (Bordeaux)"On the triviality of X0+ (pr) (Q), r>1" Let E be an elliptic curve over Q, without complex multiplication over \overline{Q}. For p a prime number, consider the representation Gal(\overline{Q} /Q )--> GL2 (Fp) induced by the Galois action on the group of p-torsion points of E. A theorem of Serre, published in 1972, asserts that there exists an integer BE such that the above representation is surjective if p is larger than BE. Serre then asked the following question: can BE be chosen independently of E? This boils down to proving the triviality, for large enough p, of the sets of rational points of four families of modular curves, namely X0 (p), Xsplit(p), Xnon-split(p) and XA4(p) (we say that a point of one of these curves is trivial if it is either a cusp, or the underlying isomorphism class of elliptic curves has complex multiplication over \overline{Q}). The (so-called exceptional) case of XA4(p) was ruled out by Serre. The fact that X0 (p)(Q ) is made of only cusps for p>163 is a well-known theorem of Mazur. In this talk we will discuss the case of Xsplit (p)(Q). Slightly more generally (because one has a Q-isomorphism between Xsplit (p) and X0+ (p2 )), we will in fact give a criterion for the triviality of X0+ (pr ) (Q) (with r>1), and show it is verified by a positive density of primes (satisfying explicit congruences). 10 November Nick Shepherd-Baron (Cambridge)"Perfect forms and moduli of abelian varieties" Perfect quadratic forms lead to a compactification of Ag whose geometry is particularly accessible. The ample classes are characterized by one inequality, generalizing the existence of the discriminant when g = 1. Over a field of char. zero, it is canonical (in the sense of Reid and Mori, not Shimura...). 17 November Toby Gee (Imperial)"Companion Forms" This will be a different talk to the one I gave last year - it will hopefully be a very relaxed introduction to the Fontaine-Mazur and Serre conjectures, and a discussion of some issues arising from these conjectures. 24 November Ian Grojnowski (Cambridge)"Geometric Satake for local fields of dimension 2" I'll explain the usual Satake isomorphism, its central role in the Langlands programme, and a generalisation of all this to fields like Qp((t)), C((s))((t)). Should be of interest to geometers also--- this can be respelled as theorems about the moduli of G-bundles on an algebraic surface--- "Donaldson theory". 1 December Shaun Stevens (UEA)"Supercuspidal representations of p-adic classical groups" The Local Langlands Correspondence relates the representations of the Weil-Deligne group of a locally compact non-archimedean local field to the irreducible smooth representations of general linear groups. Mostly conjecturally, there are also such correspondences for other p-adic groups. In this talk I will try to describe what this means, what's known and also some explicit constructions of representations for p-adic symplectic, orthogonal and unitary groups. 8 December Jens Marklof (Bristol)"Number Theory and Quantum Chaos" I will review recent developments in some fundamental problems in quantum chaos that have attracted the interest of number theorists. No knowledge of quantum mechanics is necessary. 15 December Denis Charles(Wisconsin)"Computing Modular Polynomials" (joint work with Kristin Lauter) We present a new probabilistic algorithm to compute modular polynomials modulo a prime. Modular polynomials parameterize pairs of isogenous elliptic curves and are useful in many aspects of computational number theory and cryptography. Our algorithm has the distinguishing feature that it does not involve the computation of Fourier coefficients of modular forms. We avoid computing the exponentially large integral coefficients by working directly modulo a prime and computing isogenies between elliptic curves via Velu's formulas. ## Summer 2004 This term, the seminar was held at Imperial College and was organised by Kevin Buzzard. #### The seminar programme for Summer 2004  28 April Dan Snaith (Imperial)"Overconvergent Siegel modular forms" 5 May Luis Dieulefait (Barcelona)"Existence of compatible families of Galois representations and the Fontaine-Mazur conjecture for elliptic curves" 12 May Bruno Kahn (Paris 7)"Birational motives" 19 May Fre Vercauteren (Bristol)"Zeta functions: the p-adic approach" 26 May Sarah Zerbes (Cambridge)"Selmer groups over p-adic Lie extensions" 2 June Adam Joyce (Imperial)"The Manin constant of modular abelian varieties" The Manin constant of a (modular) elliptic curve was introduced by Manin in a paper in the mid-70s, in which he also conjectured that it is always 1. I shall give this definition and discuss the results in the direction of proving the conjecture. I'll then generalise the definition to abelian varieties of arbitrary dimension and discuss the extensions of the above results to the general setting. 9 June Brian Conrey (AIM)"Random matrix theory and ranks of elliptic curves" 16 June Detlev Hoffman (Nottingham)"Isotropy of quadratic forms in finite and infinite dimension" 23 June Teruyoshi Yoshida (Harvard, visiting Imperial)"Non-abelian Lubin-Tate theory and Deligne-Lusztig theory" 4 August Chandrashekhar Khare (Utah/Tata)"Transcendental Galois representations" ## Spring 2004 This term, the seminar was held at Imperial College and was organised by Prof. Alexei Skorobogatov. #### The seminar programme for Spring 2004  14 January Kevin Buzzard (Imperial)"The 2-adic eigencurve at the boundary of weight space" Eigencurves are geometric objects parameterising certain modular forms. These objects were introduced by Coleman and Mazur in the mid-1990s and at the time very little was known about what they "looked like". Lloyd Kilford and I can write down equations for one of these eigencurves, and these equations are sufficiently explicit to enable us to pin down exactly what this eigencurve looks like near its boundary. In my talk I will give an introduction to the theory of eigencurves and then will explain a sketch of our result. 21 January Alexei Skorobogatov (Imperial)"Rational points on Enriques surfaces" The Enriques surfaces are cohomologically indistinguishable from rational surfaces, however unlike rational surfaces they have a non-trivial though very small fundamental group. For rational surfaces it is conjectured by Colliot-Th e and Sansuc, and is proved in many cases that the failure of the Hasse principle and weak approximation is controlled by the obstruction based on the Brauer group of the surface. In a joint work with David Harari we construct an Enriques surface over Q with an adelic point satisfying all the conditions provided by the Brauer group, but not in the closure of the set of Q-rational points. The proof uses descent to a torsor on this surface; the structure group of this torsor is a form of a (non-abelian) 1-dimensional orthogonal group. 28 January Roger Heath-Brown (Oxford)"Cayley's cubic surface" Roughly how many non-trivial primitive integer solutions does the equation 1/X0+1/X1+1/X2+1/X3=0 have, in a large cube max |Xi| < B ? On clearing the denominators this gives us a Cayley's cubic surface. Manin's conjecture predicts a growth rate of order B(log B)6. This has now been proved, by using the Universal Torsor for the surface. The latter is an affine variety in 16 dimensional space, which encodes all the relevant divisibility information. The proof entails counting points on this affine variety, for which the key tool comes from the geometry of numbers. 4 February Anthony Hayward (King's)"Congruences satisfied by Stark units" "Refined abelian Stark conjectures" are special value conjectures for equivariant L-functions associated to Galois extensions of global fields at "s=0". There has been a proliferation of such conjectures since Stark's original work in the 1970s, notably those of Rubin and Popescu, who generalised Stark's abelian conjecture to account for differing orders of vanishing of the L-functions, and Gross, who gave congruences for the value of the L-function. These conjectures admit a natural refinement, due to Burns, which gives Gross-style congruences in the situation of Rubin's conjecture. The formulation of the refinement is inspired by the Equivariant Tamagawa Conjecture, and this over-riding conjecture provides a unifying overview of the field which was previously sorely lacking. After describing this general situation in the first half of the talk, I move on to study those situations in which systems of explicit units provide proofs of the Burns refinement, hopefully giving more of a sense of what these conjectures are "about". The proof involves details on the explicit units, a study of ray-class fields, and some unusual combinatorics and integer identities. 11 February Victor Abrashkin (Durham)"Galois modules arising from Faltings's strict modules" A classical analogue of the concept of a finite flat group scheme over a complete discrete valuation ring loses all interesting properties in the equal characteristic case. In the past year Faltings proposed a modified definition of a group scheme with strict action and showed that it works perfectly in some situations. It will be explained in the talk that many known results about classification and arising Galois modules still hold for Faltings's modules. 18 February Kanetomo Sato (Nagoya)"p-adic ale Tate twists and arithmetic duality" For an algebraic variety X over a field k and a positive integer n invertible in k, the ale sheaf of n-th roots of unity and its tensor powers are called ( ale) Tate twists, and play fundamental roles both in number theory and in arithmetic geometry. In this talk, I will talk about a construction of p-adic Tate twists on regular arithmetic schemes (=schemes which are regular flat of finite type over Spec Z), and arithmetic duality theorems for p-adic Tate twists, which generalizes the classical Artin-Verdier duality theorem. 25 February Toby Gee (Imperial)"Companion forms over totally real fields" The companion forms conjecture was part of Serre's conjecture on the modularity of mod p Galois representations. This was proved in the early 90s by Gross and Coleman & Voloch. More recently Fred Diamond has conjectured extensions of these results to totally real fields; I will describe my recent progress on cases of these conjectures. 3 March Aleksandra Shlapentokh (East Carolina) "Hilbert's tenth problem and Mazur's conjectures" We discuss recent results concerning extensions of Hilbert's tenth problem to rings of integers of number fields and the field of rational numbers, and related conjectures of Mazur. 10 March R is de la Bret he (ENS Paris)"Counting points on varieties using universal torsors" We study the asymptotic order of the number of points of bounded height on certain varieties. Universal torsors turned out to be a useful tool to attack this kind of problem. We will give few examples to explain how to prove asymptotic estimations using universal torsors and tools of analytic number theory. 17 March Nikolaos Diamantis (Nottingham)"Second order cusp forms and L-functions" Second-order modular forms are functions that have recently appeared in several contexts: Eisenstein series formed with modular symbols, converse theorems of L-functions, percolation theory etc. They satisfy a functional equation that extends naturally that of the usual modular forms and their study is important for the topics that have motivated their introduction. We will discuss the ways they arise in various contexts, their classification and their L-functions. 24 March Adam Logan (Liverpool)"Heegner points on elliptic curves over real quadratic fields" Henri Darmon formulated a concrete version of a conjecture of Oda according to which it should be possible to construct rational points on elliptic curves over real quadratic fields defined over quadratic extensions of the field by integrating the Hilbert modular form associated to the curve. We present the conjecture together with some numerical evidence for it. This is joint work with Darmon. ## Autumn 2003 This term, the seminar was held on Wednesdays, between 4:15 and 5:15 pm in room 423 of KCL, and was organised by Dr David Solomon. #### The seminar programme for Autumn 2003 1/10/03 David Solomon, KCL Twisted Zeta-Functions, Stark Conjectures and Hilbert Symbols' ---------------------------------------------------------------- 8/10/03 John Cremona, Nottingham: Explicit Higher Descents on Elliptic Curves' ---------------------------------------------------------------- 15/10/03 Richard Hill, UCL: Fractional weights, Borcherds products and the Congruence Subgroup Problem' ---------------------------------------------------------------- 22/10/03 Tim Browning, Oxford: Counting rational points on singular cubic surfaces' ---------------------------------------------------------------- 29/10/03 Christian Elsholtz, Royal Holloway College: Additive decompositions of the set of primes' ---------------------------------------------------------------- 5/11/03 Daniel Delbourgo, Nottingham "Euler characteristics of elliptic curves via p-adic modular forms" ---------------------------------------------------------------- 12/11/03 Neil Dummigan, Sheffield, Critical values of tensor-product L-functions' ---------------------------------------------------------------- 19/11/03 Dan Evans, Nottingham: Harmonic analysis on higher dimensional local fields' ---------------------------------------------------------------- 26/11/03 Daniel Barsky, U Paris 13 [NOTE CHANGE OF DATE] Norms of Iwasawa series attached to totally real fields' ---------------------------------------------------------------- 3/12/03 Nigel Byott, Exeter: Hopf-Galois strucutres of field extensions' ---------------------------------------------------------------- 10/12/03 Prof. Igor Shparlinski, Macquarrie U. Euler Function: Smooth and Square' -------------------------------------------------  Further details will appear when available. #### The seminar programme, Summer 2003 This term it was held in the maths department of Imperial College and was organised by Dr Kevin Buzzard (buzzard@ic.ac.uk). #### The seminar programme ---------------------------------------------------------------------- 30 April --- TWO TALKS 14:15 Elmar Grosse-Kloenne (Muenster) "On twisted unit root L-functions ^^^^^ of families of varieties over finite fields" 15:45 Steve Gelbart (Weizmann) "On lower bounds for automorphic ^^^^^ L-functions". ---------------------------------------------------------------------- 7 May Allan Lauder (Oxford) "Deformation theory and the computation of zeta functions" ---------------------------------------------------------------------- 14 May Vic Snaith (Southampton) "On the Kummer--Vandiver conjecture" ---------------------------------------------------------------------- ***Tuesday 20th May*** at 1600 Victor Rotger (Barcelona)---"Diophantine properties of fake elliptic curves and their moduli spaces" ---------------------------------------------------------------------- 21 May---Ed Nevens (Imperial)---TBA (something about moduli space of abelian varieties and/or canonical subgroups, perhaps) ---------------------------------------------------------------------- 28 May---Neil Strickland (Sheffield)---"Elliptic cohomology" ----------------------------------------------------------------------  #### The seminar programme, Spring 2003 This term the seminar was held in the maths department of Imperial College on Wednesdays at 4.15 pm and was organised by Dr Kevin Buzzard (buzzard@ic.ac.uk). #### The seminar programme ---------------------------------------------------------------------- 15/1 Andrei Yafaev, Imperial Title: Descent on certain Shimura curves' Abstract: This is a joint work with Alexei Skorobogatov. Applying descent to certain unramified coverings of Shimura curves we offer an explicit method of constructing Shimura curves that do not satisfy Hasse principle; the failure of the Hasse principle is being explained by the Manin obstruction. ---------------------------------------------------------------------- 22/1 Kevin Buzzard, Imperial Title: Overconvergent 2-adic modular forms' Abstract: This is joint work with Frank Calegari. Some computations I did (and some known conjectures and theorems) led me to believe that in some cases there are very precise formulae for the p-adic valuations of the eigenvalues of T_p on various spaces of modular forms. Calegari and I have made these conjectures completely explicit and precise in the case p=2 and N=1 (for any weight k) and can prove them in some cases using a combination of deep theorems of Coleman and elementary combinatorial results involving hypergeometric function identities. ---------------------------------------------------------------------- 29/1 I. Tomasic (Leeds) "Weil conjectures--with a DIFFERENCE" ---------------------------------------------------------------------- 5/2 A. Hayward (Kings) "A conjectural class-number formula for higher derivatives of abelian L-functions" ---------------------------------------------------------------------- 12/2 R. Kucera (Brno) "Cyclotomic units" ---------------------------------------------------------------------- 19/2 (1430) J. Nekovar (Jussieu) ^^^^^^^^^^^ "On the parity of ranks of Selmer groups associated to Hilbert modular forms" 19/2 (1615) T. Ochiai (Tokyo) ^^^^^^^^^^^ "Results and examples for Iwasawa theory on Hida deformations." ---------------------------------------------------------------------- 26/2 H. Narita (Tokyo) "Fourier-Jacobi expansion of certain automorphic forms on Sp(1,q)" ---------------------------------------------------------------------- 5/3 M. Breuning (Kings) TBA (something about local epsilon constants) ---------------------------------------------------------------------- 12/3 D. Harari (Strasbourg) "Arithmetic duality theorems for 1-motives" ---------------------------------------------------------------------- 19/3 N. Broberg (Durham) "Counting rational points on finite covers of the projective plane" ---------------------------------------------------------------------- 9/4 1400: A. Yakovlev ^^^^ "Multiplicative Galois modules in local fields" 1530: G. Henniart ^^^^ "Expliciting the Langlands conjecture:the tame case" ------------------------------------------------------------------------- #### The seminar programme, Autumn 2002 This term, the seminar was held on Wednesdays, in room 423 of KCL, and was organised by Dr David Solomon. Also on Wednesdays in King's this term was the London Number Theory Study Group, which met from 2:45 to 3:45 pm.  The seminar programme 9/10 FIRST MEETING: EXCEPTIONALLY A DOUBLE-HEADER STARTING AT 2:45 in room 436: 2:45 - 3:45 (room 436) Amnon Besser, Ben Gurion University, 4:15 - 5:15 (room 423) Takao Yamazaki, Tsukuba University, Title: On the structure of Chow groups of surfaces over local fields' Abstract: Let X be a surface over a p-adic field with good reduction and let Y be its special fiber. We consider the structure of the Chow group CH0(X) of zero-cycles on X. If we write T(X) for the kernel of the Albanese map of X, then the structure of the quotient group CH0(X)/T(X) is well understood. Hence we only have to study T(X). Let D(X) be the maximal divisible subgroup of T(X). Then, it is conjectured that F(X) = T(X)/D(X) is finite and that F(X) is isomorphic to the Albanese kernel T(Y) of Y modulo p-primary torsion. On the contrary, we shall show that the p-primary torsion subgroup of F(X) can be arbitrary large even though we fix the special fiber Y. ---------------------------------------------------------------------- 16/10 Mohammed Saidi, University of Durham Title: On the fundamental group of complete curves in positive characteristics' ---------------------------------------------------------------------- 30/10 Richard Hill, UCL Title: Shintani Cocyles on GL_n' ------------------------------------------------------------------------ 6/11 No Seminar (Reading Week) ------------------------------------------------------------------------ 13/11 Sey Yoon Kim, KCL Title: On the Equivariant Tamagawa Number Conjecture for certain Quaternion Fields ' Abstract: Let L/K be a finite extension of number fields. Then the equivariant Tamagawa number conjecture relates the values of the Artin L-functions of L/K at integers to various algebraic data of L/K, and in particular, the conjecture at s=0,1 implies Chinburg's root number conjecture for L/K. We explain the conjecture at s=0 for abelian extensions over Q; then prove it for a family of biquadratic abelian extensions over Q to lift a 1989 result of Chinburg on his conjecture for the case of quaternion extensions over Q. ------------------------------------------------------------------------- 20/11 Martin Taylor, U.M.I.S.T. Title: Arithmetic Euler Characteristics' Abstract: I shall start by recalling the basic theory and constructions for Euler characteristics of varieties which support an action by a finite group. These ideas then extend firstly to the construction of equivariant Euler characteristics of arithmetic varieties, and then more generally to Euler characteristics which take into account metrics and signatures. ------------------------------------------------------------------------- 27/11 Robin Chapman, Exeter Title: Hermitian structures on lattices' Abstract: We consider lattices in Euclidean space Rn. If n is even, Rn can be given a structure of a complex vector space in many ways. Given a lattice L we investigate which C-structures on Rn have a Hermitian form compatible with the Euclidean structure on Rn and for which L becomes an O-module for some quadratic order O. In some cases we determine explicitly the O-module structure of L. ------------------------------------------------------------------------- 4/12 David Burns, King's College London Title: Nearly perfect complexes and Weil-etale cohomology' Abstract: We describe a more conceptual approach to the construction of Euler characteristics of nearly perfect complexes which was recently introduced by Chinburg, Kolster, Pappas and Snaith. We then discuss certain applications of our approach in the context of Lichtenbaum's theory of Weil- ale cohomology. ------------------------------------------------------------------------- 11/12 Rob de Jeu, Durham Title: Zagier's conjecture and (p-adic) regulators' Abstract: Let k be a number field. There is a classical relation between the residue of the zeta function of k, zetak(s), at s=1, and the regulator of the group of units of its ring of integers. Borel proved a similar relation between zetak(n) and K2n-1(k) for n>=2. The K-groups are difficult to describe explicitly. We discuss a conjecture of Zagier on how this could be done, and describe Borel's regulator (as well as a p-adic regulator) in this context, involving polylogarithms.  #### The seminar programme, Summer 2002 This term it was held in the Maths department of Imperial College and organised by Dr Kevin Buzzard (buzzard@ic.ac.uk). 24 April Frazer Jarvis (Sheffield) "Points on Fermat curves over real quadratic fields" 1 May Jayanta Manoharmayum (Sheffield) "modularity of GL2(F7) Galois representations" *2 May* Helena Verrill (Hannover) "Transportable modular symbols" 8 May Alexei Skorobogatov (Imperial) "Some new cases of the Hasse principle and weak approximation" 15 May Dan Jacobs (Imperial) Slopes of Compact Operators 22 May Ben Green (Cambridge) "Counting sumfree sets in abelian groups" 29 May Lloyd Kilford (Imperial) "Slopes of overconvergent 2-adic modular forms" 5 June Denis Petrequin (Cambridge) "Chern classes and cycle classes in rigid cohomology" 12 June Oliver Bltel (Heidelberg) TBA *13 June* Oliver Bueltel (Heidelberg) TBA (continued). 19 June Chad Schoen (Duke) "Torsion in the Chow group"  #### The seminar programme, Spring 2002 This term it was held in the Maths department of Imperial College and organised by Dr Kevin Buzzard (buzzard@ic.ac.uk).  Jan 16 Andrei Yafaev (Imperial) "Galois orbits of abelian varieties with complex multiplication Jan 23 Denis Benois (Bordeaux) "On Tamagawa numbers of crystalline representations" Jan 30 Tony Scholl (Cambridge) "Local epsilon-factors and tensor products of representations of GL(2)" Feb 6 Richard Hill (UCL) "something to do with metaplectic groups" Feb 13 John Coates (Cambridge) "Iwasawa algebras and arithmetic" Feb 20 Tim Dokchitser (Durham) "TBA" Feb 27 Susan Howson (Nottingham) "Applications of Euler Characteristics to non-Abelian Iwasawa Theory Mar 6 Shaun Stevens (Oxford) "TBA" Mar 13 John Wilson (Oxford) "Abelian surfaces with real multiplication" Mar 29 David Solomon (Kings) "Abelian Stark Conjectures in Z_p-extensions"  #### The seminar programme, Autumn 2001 This term, the seminar was held in the Maths department of Imperial College and organised by Dr Alexei Skorobogatov (a.skorobogatov@ic.ac.uk).  Oct 10 Andrei Yafaev (Imperial) "Special points on Shimura varieties" Abstract A conjecture of Andre and Oort predicts that irreducible components of a Zariski closure of a set of special points in a Shimura variety are subvarieties of Hodge type. This talk is devoted to a recent result towards this conjecture obtained in a joint work with Bas Edixhoven. Oct 17 Sir Peter Swinnerton-Dyer (Cambridge) "Rational points on certain Kummer surfaces" Abstract Most of this seminar represents joint work with Alexei Skorobogatov. Let E_1,E_2 be elliptic curves defined over an algebraic number field k, and let F_i:y_i^2=f_i(x_i) with f_i quartic be a 2-covering of E_i. Then V:y^2=f_1(x_1)f_2(x_2) is a Kummer surface associated with the Abelian surface E_1\times E_2. In the special case when E_1 and E_2 have all their 2-division points defined over k, I shall show that the Hasse Principle holds for V provided that (i) the Tate-Safarevi\v{c} groups of all the twists of E_1 and E_2 are finite, (ii) a certain rather weak technical condition holds. Here (i) is necessitated by the method of proof, but it is generally believed to be true; (ii) can be shown to be strictly stronger than the true necessary and sufficient condition, which is conjectured zto be the absence of a Brauer-Manin obstruction. Note that Schinzel's Hypothesis does not appear. The methods used have much in common with those used for diagonal cubic surfaces a_0X_0^3+a_1X_1^3+a_2X_2^3+a_3X_3^3=0 but some stages of the argument for the latter are much more complicated. Comparisons will be made between the two. Oct 24 Michael Spiess (Nottingham) "Monodromy modules and derivatives of p-adic L-functions" Oct 31 Jonathan Dee (Imperial) "Phi-Gamma-modules and families of p-adic Galois representations" Nov 7 Jean-Louis Colliot-Th e (Orsay) "Linear algebraic groups over two-dimensional fields" Abstract Let$k$be an algebraically closed field of characteristic zero. Let$K$be either a function field in two variables over$k$or the fraction field of a$2$-dimensional, excellent, strictly henselian local domain with residue field$k$. We show that linear algebraic groups over such a field$K$satisfy properties which are familar in the context of number fields: finiteness of$R\$-equivalence, Hasse principle
forprincipal homogeneous spaces of simply connected
groups,Hasse principle for complete homogeneous spaces.
This is joint work with P. Gille (Orsay) and R. Parimala
(Mumbai).

Nov 14 Kevin Buzzard (Imperial)

"The eigenvariety"

Abstract:

For a general reductive group, people are beginning to
believe that certain classes of automorphic forms on this
group lie naturally in p-adic analytic families, as the
weight varies. For GL_1 one can formulate a precise statement
and its proof is an easy consequence of global class field
theory. For GL_2 over Q, Coleman and Mazur have constructed
families interpolating classical holomorphic modular forms,
and have gone onto construct a geometric object, the
eigencurve, parameterising the forms. I will explain that if
one is willing to do a little rigid geometry then one can
generalise much of the Coleman-Mazur construction to a much
wider setting, and hence construct "the eigenvariety" in much
greater generality.

Nov 21 Vic Snaith (Southampton)

"Relative K_0, Fitting ideals and the Stickelberger phenomenon"

Nov 28 Burt Totaro (Cambridge)

"Rational points on homogeneous spaces, and the group E_8"

Abstract: A homogeneous variety over a field need not have a
rational point over the same field. The simplest example is a
conic curve, which in general has a rational point only over
a quadratic extension field. More generally, given a
semisimple group G, and a homogeneous G-variety over an
arbitrary field, we can ask what degree of field extension we
need in order to find a rational point. I will explain what
and the exceptional groups such as E_8.

Dec  5 Otmar Venjakob (Cambridge)

"Iwasawa theory of p-adic Lie extensions"

Abstract:
The most prominent example of a (non-abelian) p-adic Lie
extension K of a number field k arises maybe by adjoining to
k the p-power division points of an elliptic curve E (over k)
without complex conjugation. If G denotes the Galois group of
K/k one can study the (Pontryagin dual of the) Selmer group
of E over K as a module over the completed group algebra R(G)
of G. In this situation it is also reasonable - though not at
all obvious - to speak about pseudonull modules. One basic
result is that the Selmer group does not contain any nonzero
pseudo-null submodule (under certain conditions). If there is
enough time we are going to discuss also some features of the
general structure theory of torsion R(G)-modules up to
pseudo-isomorphism, which was proven by Coates, Schneider and
Sujatha recently.

Dec 12 No Talk



Summer 2001

This term, the seminar was held on Wednesdays in room 423 of KCL, and was organised by Dr David Solomon.

 Jun 27 Stephen Lichtenbaum (Brown University, USA)   -   "A Search for a New Cohomology Theory for Schemes over Z" (Final Seminar) Jun 26 Frank Calegari   -  unofficial' seminar at 3.30pm in room 413 in Imperial College -   Semistable Abelian Varieties of small conductor' Abstract: Fontaine proved in 1985 that there do not exist any Abelian varieties over Z. Jacobians of the modular curves X_0(p^n) (non-zero for sufficiently large n) provide examples of Abelian varieties over Z[1/p] for each p. If, however, we restrict our attention to semistable Abelian varieties,then combining Fontaine's theorem, recent papers of Brumer-Kramer and of Schoof, and some new ideas and results, we prove the following: There exists a semistable Abelian variety over Z[1/n] with n squarefree if and only if n is not in the set: {1,2,3,5,6,7,10,13}. Jun 26 William Stein   -  unofficial' seminar at 2pm in the Clore lecture theatre in Imperial College -   Mordell-Weil groups of elliptic curves over Q are visible' Abstract: During the past few years, Barry Mazur, myself, and others have studied visible subgroups of Shafarevich-Tate groups of abelian varieties. Recently, I've been studying visibility of Mordell-Weil groups of abelian varieties. In this talk, I will very briefly review some results about visibility of Shafarevich-Tate groups, then discuss some of what I've been able to prove about their counterparts in the context of visibility of Mordell-Weil groups. In particular, I will show that Mordell-Weil groups of elliptic curves over Q are visible in modular abelian varieties. Jun 15 Helena Verrill   -  unofficial' seminar at 3pm in room 413 in Imperial College -   Modular symbols computations and a certain intermediate Jacobian' Jun 13 Romyar Sharifi   -   Galois Module Structure of Local Unit Groups and Conductors' Abstract: We give an explicit description of generators of the ith unit groups of K = Qpzetapn as Galois submodules of the multiplicative group of K. We can use this to determine the ramification groups of degree pn Kummer extensions of K which are Galois over Qp. Jun 6 Richard Hill (University College, London)   -   "The Kubota Symbol on SLn" May 30 Prof. V. Nikulin (Liverpool)   -   "Dynamics of the automorphism group of a K3 surface." May 23 Daniel Delbourgo (Nottingham)   -   "Two-variable Euler systems in the non-CM case" My 16 Rob de Jeu (Durham)   -   "Algebraic K-theory and Syntomic Regulators" (provisional title)

Spring 2001

This term it was held in the maths department of Imperial College, and was organised by Dr Kevin Buzzard (k.buzzard@ic.ac.uk).

 Jan 17 Kevin Buzzard (Imperial)   -   An overview. The first talk in the study group on "Remarks on a conjecture of Fontaine and Mazur" by R. Taylor Jan 24 Al Weiss (U Alberta) "Equivariant Iwasawa Theory" Jan 31 Andreas Langer (Bielefeld) "Relative de Rham-Witt cohomology" Feb 7 Colin Bushnell (Kings) "Local Langlands correspondences: some explicit problems" Feb 14 Alexei Skorobogatov (Imperial) "Rational solutions of certain equations involving norms" Feb 21 Christophe Cornut (Strasbourg) "A conjecture of B. Mazur on higher Heegner points" Feb 28 Keith Ball (UCL) "There are infinitely many irrational values of zeta at the odd numbers" Mar 7 Werner Hoffman (Humboldt U) "On the differential equations satisfied by weighted orbital integrals" Mar 14 Anupam Saikia (Cambridge) Title TBA Mar 21 Victor Flynn (Liverpool) "Coverings of curves" Abstract: We shall discuss the idea of finding all rational points on a curve C by first finding an associated collection of curves whose rational points cover those of C. This classical technique has recently been given a new lease of life by being combined with descent techniques on Jacobians of curves and Chabauty techniques. We shall survey recent applications during the last 5 years which have used Chabauty techniques and covering collections of curves of genus 2 obtained from pullbacks along isogenies on their Jacobians.

Autumn 2000

This term it was held in the maths department of Imperial College, and was organised by Dr Alexei Skorobogatov (a.skorobogatov@ic.ac.uk).

 Oct 11 Kevin Buzzard (IC) "Questions about slopes of modular forms" Oct 18 Alexei Skorobogatov (Imperial) "Introduction to the study group: McCallum's work on the Brauer points on Fermat curves" Oct 25 David Solomon (KCL) "A p-adic Stark conjecture in Zp-extensions" Nov 1 Kevin Buzzard (IC) "Continuation of the study group on McCallum's work: Coleman's integral and applications to rational points" Nov 8 Jan Nekovar (Cambridge) "Some consequences of Mazur's conjecture on Heegner points" Nov 15 Tom Fisher (Cambridge) "The Cassels-Tate pairing and the icosahedron" Nov 22 Roger Heath-Brown (Oxford) Counting rational points on algebraic curves and surfaces; Nov 29 Jean-Robert Belliard (Nottingham) "Real analogues of Stickelberger's theorem" Dec 6 Richard Hill (UCL) "Syntomic regulators and K-theory of curves"

Summer 2000

This term, the seminar was held in KCL, and was be organised by Dr David Burns.  May 24 Richard Hill (UCL) "Metacyclic Covers of GLn" May 31 Frazer Jarvis (University of Sheffield) "Fermat's last theorem over totally real fields?" Jun 7 Alexei Skorobogatov (ICL) "Swinnerton-Dyer's theorem on diagonal cubic surfaces" n.b. this talk starts at 5pm Jun 14 Anton Deitmar (Exeter) "Geometric zeta functions and class numbers of cubic orders" Jun 21 No seminar scheduled Jun 27 Cornelius Greither   (U. der Bundeswehr, Munich) "TBA" n.b. this is a Tuesday, from 4.30 to 5.30 pm

Spring 2000

This semester, the seminar was held on Wednesdays, in room 807 of UCL, and was organised by Dr Richard Hill.
The programme was:

 Jan 26 Neil Dummigan (Oxford) "Some evidence for the Bloch-Kato conjecture at critical points" Feb  2 Susan Howson (Nottingham) "non-Abelian Iwasawa theory and applications to elliptic curves" Feb  9 David Burns (KCL) "On the conjectures of Gross and Tate concerning values of abelian L-functions at s = 0" (This will be the first talk in a study group) Feb 16 Victor Abrashkin "Ramification Filtrations and a local analogue of the Grothendieck Conjecture" Feb 23 David Burns (KCL) "Continuation of the study group on values of Abelian L-functions at s=0" Mar 1 Richard Hill Continuation of the study group. Mar 8 Robert Vaughan "A Generalized Divisor Problem". Mar 15 David Solomon Continuation of the study group. Mar 22 David Solomon Continuation of the study group on values of Abelian L-functions at s=0.

Autumn 1999

For the first three meetings of this semester, the seminar was held on Wednesdays at King's, and was organised by Dr David Solomon.

From the 4th of November, the seminars moved to Imperial College and were organised by Dr Kevin Buzzard.

Weds Oct 13, KCL
Bart de Smit (Leiden, visiting KCL)
"On Imin Chen's jacobian relations"
ABSTRACT:
This is joint work with Bas Edixhoven. We generalise a result of Chen concerning an isogeny between products of jacobians of modular curves associated to subgroups of GL(2,p)

Weds Oct 20, KCL
Olaf Neisse (Augsburg, visiting Southampton)
"On balanced modules of group algebras"

Weds Oct 27, KCL
David Solomon (KCL)
"Cyclotomic p-Units: a Wild Euler System"

Thurs Nov 4, room 6M42, IC
Kevin Buzzard (IC)
"Some new cases of a conjecture of Artin."
This talk will be followed by a short meeting to organise the remainder of this term's seminars. Suggestions include research talks, a study group (a recent preprint of Heath-Brown has been suggested), or a mixture of the two.

Thurs Nov 11,
 Cambridge-Oxford-Warwick (COW) algebraic geometry seminar, room 642 IC 2.00 Nick Shepherd-Barron (Cambridge CMS) - Isomorphism classes in local moduli spaces 3.15 Paul Seidel - Floer cohomology and the Picard-Lefschetz formula
Thurs Nov 11, 4.30 pm, room 642 IC
Alexei Skorobogatov
"Rational points on diagonal cubic surfaces"
This is the first talk on what will be a short study group on a short paper of Heath-Brown.

Thurs Nov 18, 4.00 pm, room 6M42 IC
Kevin Buzzard
"Some new cases of Artin's conjecture"
Thurs Nov 25, 4.00 pm, room 6M42 IC
Alexei Skorobogatov
"more about Heath-Brown's paper on cubic surfaces"
Thurs Dec 2, 4.00 pm, room 6M42 IC
Richard Hill
"The next installment in Heath-Brown's paper on cubic surfaces"

The Seminar was held weekly in Summer 1999 and comprised a working group on Rubin's article:
The work of Kolyvagin on the Arithmetic of Elliptic Curves'
in The arithmetic of complex manifolds, SLN 1399, (1989) p.128-136

May   5    David Solomon (KCL)
Introductory talk

May 19    David Solomon (KCL)
Second introductory talk

May 19    Richard Hill
Rubin's proposition 1 and the Tate pairing'

Jun   3    Richard Hill
Second talk

Jun   9    Kevin Buzzard
First talk

Jun   16    Kevin Buzzard
Second talk

Jun   23    David Burns
Equivariant forms of the Birch-Swinnerton Dyer Conjecture

In the Spring of 1999, the seminar was held in the maths department of Imperial College and was organised by Dr Kevin Buzzard (buzzard@ic.ac.uk).

Mar 22 1999    Ivan Fesenko (Nottingham)
Local reciprocity cycles'

Mar 15 1999    Bernhard Koeck (Karlsruhe)
Computing the homology of Koszul complexes

Mar  8 1999    Kevin Buzzard
Euler Systems: the main theorems

Mar  1 1999    David Solomon (Kings)
p-adic Stark Conjectures at s=1: theory and computation II
- the computation

Feb 22 1999    Richard Hill
Examples of Selmer groups' (study group)
(this talk will be inspired by Rubin's forthcoming book)

Feb 15 1999    David Solomon (Kings)
p-adic Stark Conjectures at s=1: theory and computation

Feb  8 1999    Kevin Buzzard (Imperial)
Tate modules and etale cohomology (study group) sheaves will be assumed)

Feb  1 1999    John Dee (Cambridge)
Chow Groups of Self Products of CM Elliptic Curves

Jan 18 1999    Richard Hill
Geometry of Numbers and Class Field Theory

In the Autumn of 1998 it was held on Fridays, from 11:15 to 12:15 in Room 807 of the UCL Maths department, and was organised by Dr. Richard Hill (rih@math.ucl.ac.uk) of UCL.

Dec 11 1998   David Solomon
Dedekind Sums and Cocycles on GLn(Q)

Abstract:
In 1989 Glenn Stevens showed how the periods' of Eisenstein series could be used to define certain families of 1-cocycles on GL2(Q) whose values can be expressed in terms of Dedekind Sums (for example, those appearing in the transformation formula for Dedekind's eta-function). In fact, the cocycle relation is equivalent to (a generalisation of) the classical Dedekind Reciprocity Law for these sums. Stevens also showed how these cocycles could be used to evaluate the partial zeta-functions of real quadratic fields at non-positive integers. In 1992, Robert Sczech constructed 1-cocycles on GL2(Q) by a different method, using only real analysis. He too showed how they related to Dedekind Sums and partial zeta-values. (Indeed they are very closely related to Stevens'). In 1993 Sczech extended his construction to GLn(Q), producing (n-1)-cocycles that are similarly universal' for the evaluation of partial zeta-values over totally real fields of degree n.

In this talk, I shall present a third construction of cocycles on GLn(Q) for n=2 and 3, which was inspired by Shintani's formulae for partial zeta-values. They are again closely related to Stevens' and Sczech's but the construction is algebraic and elementary. I shall also explain the connections with Dedekind Sums and mention (time allowing) partial zeta-values, p-adic interpolation, and the challenge of higher dimensions'.

Dec 4 1998   Alexei Skorobogatov
Second cohomology and the existence of rational points

Abstract:
Let S be an algebraic group over a field k, and X be a k-variety. We denote by X' the same variety considered over the algebraic closure of k. If Y'/X' is a torsor under S (equipped with a suitable Galois action), then the obstruction for Y'/X' to come from some Y/X (defined over k) lies in the second cohomology set of S. If X contains a k-point, then this class is neutral. In some arithmetically meaningful cases, e.g. X a principal homogeneous space of a semi-simple group over a totally imaginary number field k, and Y'/X' is the universal covering, the converse is also true. Using these ideas one can give a short proof of an old theorem of Sansuc that the Manin obstruction is the only obstruction to the Hasse principle for principal homogeneous spaces of semi- simple groups over number fields (here S is Abelian).

Nov 27 1998    No number theory seminar
(UCL was hosting a one-day event: A tribute to the life and work of Sir James Lighthill')

Nov 20 1998    Jan Nekovar
A skew-symmetric pairing on the infinite part of a Selmer group

Abstract:
The talk will be about a non-classical generalisation of the Cassels-Tate pairing. It is defined, roughly speaking, on the non-generic part of the dual of the Selmer group. The existence of the pairing has non-trivial consequences; for example, one can deduce results about the parity of the (co)-rank of the Selmer group.

Nov 13 1998    Kevin Buzzard
Families of modular forms and the Gouvea-Mazur conjecture

Abstract:
Recent deep work of Coleman has shown what people have suspected now for a long time - namely that many modular forms come in families'. Coleman's work establishes (a slightly weak form of) a conjecture of Gouvea and Mazur. I will explain the conjecture and show how one can use a completely different (and much simpler method) to attack it. The simpler method does not give results as strong as Coleman's (one only gets continuity' results rather than `analyticity' results), but has the advantage that it generalises much more easily to other kinds of modular forms.