London Number Theory Seminar Previous Seminars 


08/01/18 to 11/01/18: UKJapan Winter School 2018 on Number Theory  Galois representations and Automorphic Forms.
10/01/18 Martin Widmer (Royal Holloway)
Title: Average bounds for the $\ell$torsion in class groups of number fields
Abstract: Let $\ell$ and $d$ be integers >1. By the CohenLenstraMartinet heuristics the $\ell$torsion part of the class groups of degree $d$ number fields should be "very small" in terms of the discriminant for "almost all" such fields. However, nontrivial such bounds for all $\ell$ are known only for $d\leq 5$ due to recent work of Ellenberg, Pierce, and Wood. We explain their strategy, how one can improve their bounds for $d=4,5$, and we also present analogous results for certain
families of arbitrarily large degree. (This is joint work with Christopher Frei).
17/01/18 Kaisa Matomaki (University of Turku)
Title: Multiplicative functions in short intervals revisited
Abstract: A couple of years ago Maksym Radziwill and I showed that the average of a multiplicative function in almost all very short intervals is close to its average on long intervals. This result has found many applications. In a work in progress that I will talk about, Radziwill and I revisit the problem and generalise the result to functions which vanish often as well as improve on the upper bound for the number of possible exceptional intervals. This new work has applications for instance to the gaps between numbers that can be represented as a sum of two squares.
24/01/18 Andrea Ferraguti (University of Cambridge)
Title: Strongly modular models of $\mathbb{Q}$curves
Abstract: A strongly modular $\mathbb{Q}$curve is a nonCM elliptic curve over a number field whose Lfunction is a product of Lfunctions of classical weight 2 newforms. We address the problem of deciding when an elliptic curve has a strongly modular model, showing that this holds precisely when the curve has a model that is completely defined over an abelian number field. The proof relies on Galois cohomological methods. When the curve is defined over a quadratic or biquadratic field, we show how to find all of its strongly modular twists, using exclusively the arithmetic of the base field. This is joint work with Peter Bruin.
31/01/18 Samir Siksek (University of Warwick)
Title: Frey curves, short character sums and a problem of Erdős
Abstract: Consider the following Diophantine problem:
\[
n(n+d)(n+2d)\cdots (n+(k1)d)=y^\ell, \qquad \gcd(n,d)=1,
\]
where $n$, $d$, $y$ are integers and the exponent $\ell$ is prime.
There are obvious solutions with $y=0$ or $d=0$. A longstanding
conjecture of Erdős states that if $k$ is suitably large then the
only solutions are the obvious ones. We show that if $k$ is suitably
large then either the solution is one of the obvious ones, or $\ell<
\exp(10^k)$.
Our methods include Frey curves and Galois representations,
the prime number theorem for Dirichlet characters, results on
exceptional zeros of Dirichlet Lfunctions, the large sieve, and
Rothlike theorems on the existence of 3term arithmetic progressions
in certain sets. This is joint work with Mike Bennett.
07/02/18 Sam Chow (University of York)
Title. Bohr sets and multiplicative diophantine approximation
Abstract. In two dimensions, Gallagher's theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fibre version of Gallagher's theorem, sharpening and making unconditional a result recently obtained conditionally by Beresnevich, Haynes and Velani. The idea is to find large generalised arithmetic progressions within inhomogeneous Bohr sets, extending a construction given by Tao. This precise structure enables us to verify the hypotheses of the DuffinSchaeffer theorem for the problem at hand, via the geometry of numbers.
14/02/18 Valentina Di Proietto (University of Exeter)
Title: A nonabelian algebraic criterion for good reduction of curves
Abstract: For a family of proper hyperbolic complex curves
$f: X \longrightarrow \Delta^*$ over a puntured disc $\Delta^*$ with semistable reduction at the
center, Oda proved, with transcendental methods, that the outer
monodromy action of $\pi_1(\Delta^*) \cong \mathbb{Z}$ on the classical unipotent
fundamental group of the generic fiber of $f$ is trivial if and only if f has
good reduction at the center. In this talk I explain a joint project
with B. Chiarellotto and A. Shiho in which we give a purely algebraic
proof of Oda's result.
21/02/18 Ildar Gaisin (École polytechnique)
Title: Fargues' conjecture in the GL_2case.
Abstract: Recently Fargues announced a conjecture which attempts to geometrize the (classical) local Langlands correspondence. Just as in the geometric Langlands story, there is a stack of Gbundles and a Hecke stack which one can define. The conjecture is based on some conjectural objects, however for a cuspidal Langlands parameter and a minuscule cocharacter, we can define every object in the conjecture, assuming only the local Langlands correspondence. We study the geometry of the nonsemistable locus in the Hecke stack and as an application we will show the Hecke eigensheaf property of Fargues conjecture holds in the GL_2case and a cuspidal Langlands parameter. This is joint work with Naoki Imai.
28/02/18 Edva RodittyGershon (University of Bristol)
Title: Arithmetic statistics of higher degree Lfunctions over function fields.
Abstract: A classical problem in number theory is to evaluate the number of primes in an arithmetic progression. This problem can be formulated in terms of the von Mangoldt function. I will introduce some conjectures concerning the fluctuations of the von Mangoldt function in arithmetic progressions. I will also introduce an analogous problem in the function field setting and discuss its generalization to arithmetic functions associated with higher degree Lfunctions (in the limit of large field size). The main example we will discuss is an elliptic curve Lfunction and statistics associated with its coefficients. This is a joint work with Chris Hall and Jon Keating.
07/03/18 Eyal Goren (McGill University)
Title: Theta operators for unitary modular forms.
Abstract: This is joint work with Ehud De Shalit (Hebrew University). We shall consider padic modular forms on a unitary Shimura variety associated to a quadratic imaginary field, where p is inert in the field, and its mod p reduction. In this case, theta operators were constructed by Eischen and Mantovan, and by De ShalitGoren, independently and using different approaches. I will describe our approach that makes heavy use of Igusa varieties. The main two theorems are a formula for the effect of a theta operator on qexpansions and its analytic continuation from the ordinary locus to the whole Shimura variety in characteristic p. Along the way interesting questions about filtrations of automorphic vector bundles arise and, to the extent time allows, I will discuss these questions in light of our work on foliations on unitary Shimura varieties.
14/03/18 Daniel Disegni (Université ParisSud)
Title: The universal $p$adic GrossZagier formula.
Abstract: Around 1986 three great theorems were proved: Gross and Zagier related Heegner points on elliptic curves to derivatives of Lfunctions; Waldspurger related toric periods of automorphic forms to special Lvalues; and Hida showed that ordinary modular forms and their Lfunctions vary in $p$adic families.
I will explain how the spirits of the second and third results can be infused into the first one. The outcome is a $p$adic GrossZagier formula, valid for Hida families of cuspforms for the group $GL(2)\times U(1)$ over a totally real field. Combined with work of Fouquet, it has applications to the $p$adic BlochKato conjecture for the Selmer groups of such forms.
21/03/18 Two seminars: Vincent Pilloni (ENS Lyon) and Peter Schneider (University of Muenster).
Schneider (15301530): Character varieties and $(\varphi_L,\Gamma_L)$modules.
Abstract: After reviewing old work with Teitelbaum, in which we constructed the character variety $X$ of the additive group $o_L$ in a finite extension $L/Q_p$ and established the Fourier isomorphism for the distribution algebra of $o_L$, I will briefly report on more recent work with Berger and Xie, in which we establish the theory of $(\varphi_L,\Gamma_L)$modules over $X$ and related it to Galois representations. Then I will discuss an ongoing project with Venjakob. Our goal is to use this theory over $X$ for Iwasawa theory.
Pilloni (16001700): Higher padic families of Siegel modular forms
Abstract: We will describe a theory of padic families of higher coherent cohomology classes for the group GSp_4/Q together with some arithmetic applications to the HasseWeil Zeta function of genus 2 curves (joint with G. Boxer, F. Calegari, T. Gee) or to the construction of Galois representations.
4/10/17 Laura Capuano (Oxford)
Title: Unlikely intersections in families of abelian varieties and some polynomial Diophantine equations
Abstract: What makes an intersection likely or unlikely? A simple dimension count shows that two varieties of dimension $r$ and $s$ are non "likely" to intersect if $r <\,$codim $s$, unless there are some special geometrical relations among them. A series of conjectures due to BombieriMasserZannier, Zilber and Pink rely on this philosophy. After a small survey on these problems, I will speak about a joint work with F. Barroero (Basel) in this framework in the special case of curves in families of abelian varieties. This gives also applications to the study of the solvability of the so called "almostPell" equations, generalising some results due to Masser and Zannier.
11/10/17 Abhishek Saha (Queen Mary)
Title: Integral representation and critical $L$values for the standard $L$function of a Siegel modular form
Abstract: I will talk about some of my recent work with Pitale and Schmidt where we prove an explicit pullback formula that gives an integral representation for the twisted standard $L$function for a holomorphic vectorvalued Siegel cusp form of degree $n$ and arbitrary level. In contrast to all previously proved pullback formulas in this situation, our formula involves only scalarvalued functions despite being applicable to $L$functions of vectorvalued Siegel cusp forms. Further, by specializing our integral representation to the case $n = 2$, we prove an algebraicity result for the critical $L$values in that case (generalizing previously proved criticalvalue results for the full level case).
18/10/17 Francis Brown (Oxford/IHES)
Title: Nonholomorphic modular forms for $\mathrm{SL}_2(\mathbb{Z})$
Abstract: I will discuss elementary properties of a class of nonholomorphic modular forms for the full modular group $\mathrm{SL}_2(\mathbb{Z})$. It contains the classical holomorphic theory, but is completely distinct from Maass' theory of harmonic modular forms. This class is related to the theory of mixed motives, mapping class groups, mock modular forms, and string theory. I may mention one or two of these connections.
25/10/17 Eugen Hellmann (Muenster)
Title: A local model for the triangulate variety and applications to $p$adic automorphic forms
Abstract: In joint work with C. Breuil and B. Schraen we prove (under mild additional hypothesis) that $p$adic automorphic forms on eigenvarieties for definite unitary groups are classical if their associated $p$adic Galois representation is crystalline at places dividing $p$.
In the same setup we further determine all nonclassical overconvergent forms of finite slope that give rise to the same Galois representation as a classical automorphic form.
These results rely on a close analysis of the local geometry of a space parametrizing $p$adic Galois representations with a certain prescribed behaviour at $p$  so called triangulate representations. We study this geometry by proving that the space is smoothly equivalent to a certain variety showing up in geometric representation theory.
1/11/17 Judith Ludwig (Bonn)
Title: A quotient of the LubinTate tower
Abstract: In this talk I will report on joint work with Christian Johansson. The aim of our project is to construct a quotient of an infinite level LubinTate space by a certain parabolic subgroup of $\mathrm{GL}(n,F)$ ($F/ \mathbb{Q}_p$ finite) as a perfectoid space.
The motivation for constructing this quotient is as follows. As I will explain in the talk, Scholze recently constructed a candidate for the mod $p$ JacquetLanglands correspondence and the mod $p$ local Langlands correspondence for $\mathrm{GL}(n, F)$. Given a smooth admissible representation $\pi$ of $\mathrm{GL}(n, F)$, the candidate for these correspondences is given by the etale cohomology groups of the adic projective space $\mathbb{P}^{n1}$ with coefficients in a sheaf $F_\pi$ that one constructs from $\pi$.
The finer properties of this candidate remain mysterious.
As an application of the quotient construction one can show a vanishing result for some of these cohomology groups $\mathrm{H}^i_{et}(\mathbb{P}^{n1},F_\pi)$.
8/11/17 JeanFrancois Dat (Paris VI)
Title: Jordan decomposition for $\ell$blocks of $p$adic groups
Abstract: Not much is known on the $\ell$modular or $\ell$integral representation theory of $p$adic groups, beyond the case of $GL_n$. Even worse, the main property used by Vigneras in her treatment of the $GL_n$ case is now known to generally fail for other groups. Inspired by the theory of ``Jordan decomposition'' for $\ell$blocks of finite reductive groups, we have conjectured the existence of a decomposition of $\ell$integral representations into factors parametrized by Langlands parameters with source the primeto$\ell$ inertia subgroup, and which obeys some version of the Langlands functoriality principle associated to a morphism of $L$groups. I will discuss recent progress on this conjectural picture, both for depth 0 and for positive depth representations.
15/11/17 Torsten Wedhorn (Darmstadt)
Title: Variations of invariants of reductive group actions
Abstract: Whenever a reductive group acts on a variety there are certain fundamental invariants of this action: complexity,
weight lattice, valuation cone. I will explain these notions
and report on recent work how these invariants vary
in families.
22/11/17 Jack Thorne (Cambridge)
Title: Life after the LanglandsTunnell theorem
Abstract: The theorem of the title states that if $K$ is a number field, then any representation $\rho : G_K \to \mathrm{GL}_2(\mathbb{C})$ with projective image $S_4$ arises from automorphic forms.
Wiles famously used this theorem in the case $K = \mathbb{Q}$, together with a grouptheoretic coincidence, to establish the automorphy of the mod 3 representations attached to elliptic curves over $\mathbb{Q}$. The rest of the story you know.
We will discuss what goes wrong with this argument when studying elliptic curves over more general number fields, and what one can do instead.
2728/11/17 The LondonParis Number Theory Seminar (in Paris)
29/11/17 Chris Williams (Imperial)
Title: $p$adic Asai Lfunctions of Bianchi modular forms
Abstract: The Asai (or twisted tensor) Lfunction attached to a Bianchi modular form is the 'restriction to the rationals' of the standard Lfunction. Introduced by Asai in 1977, subsequent study has linked its special values to the arithmetic of the corresponding form. In this talk, I will discuss joint work with David Loeffler in which we construct a $p$adic Asai Lfunction  that is, a measure on $\mathbb{Z}_p^{*}$ that interpolates the critical values $L^{As}(f,\chi,1)$  for ordinary weight 2 Bianchi modular forms. The method makes use of techniques from the theory of Euler systems, namely Kato's system of Siegel units, building on the rationality results of Ghate.
6/12/17 Florian Herzig (Toronto)
Title: Ordinary representations and locally analytic socle for $\mathrm{GL}_n(\mathbb{Q}_p)$
Abstract: Suppose that $\rho$ is an irreducible automorphic $n$dimensional global $p$adic Galois representation that is uppertriangular locally at $p$. In previous work with Breuil we constructed a unitary representation of $\mathrm{GL}_n(\mathbb{Q}_p)$ on a $p$adic Banach space (depending only on $\rho$ locally at $p$) that is an extension of finitely many principal series, and we conjectured that this representation occurs globally in a space of $p$adic automorphic forms cut out by $\rho$. In work in progress we prove many new cases of this conjecture, assuming that $\rho$ is moreover crystalline.
13/12/17 Cong Xue (Cambridge)
Title: Cuspidal cohomology of stacks of shtukas
Abstract: We will talk about the $\ell$adic cohomology of the classifying stacks of shtukas for a constant split reductive group over a function field. We will construct the constant term morphisms on the cohomology groups. And we will show that the cuspidal cohomology, defined as the intersection of the kernels of these constant term morphisms, is of finite dimension and equals to the Heckefinite cohomology defined by V. Lafforgue. The essential ingredients are the compatibility of the geometric Satake equivalence with the constant term functor and the relative contractibility of deep enough HarderNarasimhan strata in the stacks of shtukas.
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 WeilDeligne 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.
03/05/17 Vladimir Dokchitser (KCL)
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.
17/05/17 Nadav Yesha (KCL)
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 finescale 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 GrossZagier formula for a certain anticyclotomic padic Lfunction 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 padic Lfunction L_p(E/K) in terms of distributions of Heegner points on Shimura curves that are rational over the anticyclotomic Z_pextension of K. The "special value" of L_p(E/K) is 0, and I will sketch a proof of a GrossZagier formula for the first derivative of L_p(E/K) involving a Heegner point over K and a padic Linvariant of E à la MazurTateTeitelbaum. The strategy is based on level raising arguments and Jochnowitztype congruences. This is joint work (in progress) with Rodolfo Venerucci.
31/5/17 Wushi Goldring (Stockholm Uni)
Title: Geometry engendered by GZips: Shimura varieties and beyond.
Abstract: Moonen, Pink, Wedhorn and Ziegler initiated a theory of
GZips, which is modeled on the de Rham cohomology of varieties in
characteristic p>0 "with Gstructure", where G is a connected
reductive F_pgroup. Building on their work, when X is a good
reduction special fiber of a Hodgetype Shimura variety, it has been
shown that there exists a smooth, surjective morphism \zeta from X to
a quotient stack GZip^{\mu}. When X is of PEL type, the fibers of
this morphism recover the EkedahlOort 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 GZip viewpoint shows that they hold much more generally, for geometry engendered by GZips: Any scheme Z equipped with a morphism to GZip^{\mu} satisfying some general schemetheoretic 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 twodimensional 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 DeligneRibet $p$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 Lfunctions.
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.
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 nonordinary 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, LeiLoefflerZerbes have constructed an Euler system for the tensor product of the padic 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.
25/01/17 Rachel Newton (Reading University)
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 2dimensional 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 AndreattaIovitaPilloni and LiuWanXiao on the geometry of the ColemanMazur 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)
Title: A padic JacquetLanglands correspondence
Abstract: In this talk, I will construct a padic JacquetLanglands 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 localglobal compatibility with the completed cohomology of Shimura curves, as well as a compatibility with the classical JacquetLanglands 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 padic places. It turns out that this question is more treatable. Moreover, using a product formula formula, the padic 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 MordellWeil theorem, the group of Krational points on A is finitely generated and as for elliptic curves, its rank is predicted by the Birch and SwinnertonDyer conjecture. A basic consequence of this conjecture is the parity conjecture: the sign of the functional equation of the Lseries determines the parity of the rank of A/K. Under suitable local constraints and finiteness of the ShafarevichTate 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 nonabelian 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 padic integration running 13001415 and one on padic Hodge theory running 14301600.
5/10/16 James Newton (Kings)
Title: Patching and the completed homology of locally symmetric spaces.
Abstract: I will explain a variant of TaylorWiles 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 (padic) automorphic forms. This is joint
work with Toby Gee.
12/10/16 JeanStefan 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 ptorsion 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 p1 power of the Hodge bundle which vanishes exactly on its complement. In this talk, we will explain a grouptheoretical construction of generalized Hasse invariants based on the stack of Gzips introduced by Pink, Wedhorn, Ziegler Moonen. When S is the good reduction special fiber of a Shimura variety of Hodgetype, we show that the EkedahlOort 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 Heckeeigenvalues 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 FerreroWashington theorem for $\mathcal{F}_{\mathfrak{p}}$. (Joint work with Bruno
Anglès, Francesc Bars and Ignazio Longhi)
26/10/16 Brian Conrad (Stanford)
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 localglobal 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 degree1 TateShafarevich
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 positivedimensional (possibly nonsmooth) affine algebraic $k$group scheme and its (typically nonrepresentable) ${\rm{GL}}_1$dual sheaf for the fppf topology.
2/11/16 Joe KramerMiller (UCL)
Title: Fisocrystals 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 Fisocrystal. By work of Tsuzuki and Crew an Fisocrystal is overconvergent precisely when $\rho$ has finite monodromy. However, in practice most Fisocrystals 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 Galoistheoretic interpretation of these log decay Fisocrystals 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 ppart of the Birch and SwinnertonDyer Conjecture for
Elliptic Curves of Analytic Rank One
Abstract: I will explain the recent proof of the ppart of the Birch
and SwinnertonDyer 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 HarderNarasimhan 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 pdivisible 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 nonempty.
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)
Title: padic Galois representations and padic L functions
Abstract: Let p be a prime number. We will discuss how to associate, to a modular
form f of level N, a (partial) padic Lfunction interpolating special
values of the complex Lfunction 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 padic Lfunction, and on the other hand
results on Kato's local epsilon conjecture 2dimensional 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 nonCM 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.
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 socalled 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 padic Lfunction associated with Siegel forms
Abstract: We shall begin with an introduction to Lfunctions in arithmetic, their padic interpolation and trivial zero. We shall state a conjecture of Greenberg and Benois which predicts the order and the leading coefficient of padic Lfunctions when trivial zeros appear. We shall then explain how one calculates the first derivative of the standard padic Lfunction 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 Qpsplit 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 JordanHölder factors. Using the Gequivariant 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 JordanHölder 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: Subconvexity in certain Diophantine problems via the circle method.
Abstract: The subconvexity barrier traditionally prevents one from applying the HardyLittlewood (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 squareroot 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 subconvexity 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 translationinvariant type.
6th and 7th June  LondonParis Number Theory Seminar.
8/6/16 Adam Harper (Cambridge)
Title: Gaussian and nonGaussian 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 nonprincipal 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 qprimitive points on expanding horospheres as q tends to infinity.
22/6/16 Maria Valentino (King's)
Title: On the diagonalizability of the Atkin Uoperator 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 Uoperator carrying the Hecke action over harmonic cocycles.
29/6/16 Nina Snaith (Bristol)
Title: Unearthing random matrix theory in the statistics Lfunctions: 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 Lfunctions 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 longstanding 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)
Abstract: Following Ribet's seminal 1976 paper there have been many results employing congruences between stable cuspforms and lifted forms to construct nonsplit extensions of Galois representations. This strategy can be extended to construct elements in the BlochKato 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 FontaineMazur style conjectures.
Abstract: The theory of modular curves, their integral models and modular forms on them is welldeveloped, 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, BultelWedhorn 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).
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 KohnenShintani 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 StarkHeegner points. Then, we will show how these Darmon cycles are related to (a padic family of) halfintegral weight modular forms. The relation follows by the padic interpolation of a well known formula of Waldspurger.
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, GonzalezAviles and Rubin proved that if L(E/Q,1) is nonzero, then the full BirchSwinnertonDyer 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.
Eugen Hellmann (Bonn)
Title: On companion points on eigenvareties
Abstract: In the theory of padic modular forms (or more generally padic automorphic forms) the phenomenon occurs that there are nonclassical and classical forms that have the same system of Heckeeigenvalues. This phenomenon has an explanation in terms of the associated Galois representations. Namely certain padic Hodgestructures (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.
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.
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.
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 Zetafunction.
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 BrauerManin 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 ColliotThelene about the sufficiency of the BrauerManin obstruction for varieties admitting a suitable fibration. This is joint work with Damaris Schindler.
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 GrossStark conjecture
Abstract: In 1980, Gross conjectured a formula for the expected leading term at $s=0$ of the DeligneRibet $p$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.
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 halfplane.
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 noncompact type.
14/10/15 Stephane Bijakowski (Imperial)
Title: The partial degrees of the canonical subgroup
Abstract: If the Hasse invariant of a pdivisible group is small enough, then one can construct a canonical subgroup inside its ptorsion. 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 pdivisible 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: Linvariants and localglobal compatibility for GL2
Abstract: Let F be a totally real number field, w a prime of F above p, V a 2dimensional padic 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 semistable noncrystalline in Fontaine's sense, we can associate to Vw the socalled FontaineMazur Linvariants, which are invisible in classical local Langlands correspondance. We show that these Linvariants can be found in the completed cohomology group of Shimura curves.
28/10/15 Rob Kurinczuk (Bristol)
Title: Cuspidal lmodular representations of classical padic groups
Abstract: For a classical groups (unitary, special orthogonal, symplectic) over locally compact nonarchimedean 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: pkernels occurring in isogeny classes of pdivisible groups
Abstract: I will give a criterion which allows to determine, in terms of the combinatorics of the root system A_n, which pkernels occur in a given isogeny class of pdivisible groups over an algebraically closed field of positive characteristic. This question is related to the relationship between Newton and EkedahlOort strata on reductions of Shimura varieties as well as the nonemptiness of affine DeligneLusztig varieties.
9/11/15 LondonParis Number Theory Seminar (in Paris)
11/11/15 Laurent Berger
Title: Iterated extensions and relative LubinTate groups
Abstract: An important construction in padic 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 padic dynamical systems, Coleman power series and relative LubinTate groups.
18/11/15 Julien Hauseux (Kings)
Title: Parabolic induction and extensions
Abstract: Let G be a padic 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 deltafunctor 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 padic 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 NéronOggShafarevich 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 wellknown 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 MordellWeil rank of the Jacobian of C is less than g, the
ChabautyColeman method can often be used to find these rational
points through the construction of certain padic integrals.
When the rank is equal to g, we can use the theory of padic height
pairings to produce padic 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 Endoclasses for classical groups.
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 BreuilMézard 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
padic 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)
Title: Badly approximable numbers
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: RapoportZink spaces of Hodge type and applications to Shimura varieties
Abstract: RapoportZink spaces of (P)EL type are local analogues of Shimura varieties of PEL type. Examples include LubinTate spaces and Drinfeld upper half spaces.
In this talk, we construct the "Hodgetype generalisation" of RapoportZink spaces under the unramifiedness hypothesis, and apply it to the integral models of Hodgetype Shimura varieties. The new examples are Spin and orthogonal RapoportZink spaces (of arbitrary rank)  local analogues of Spin and orthogonal Shimura varieties.
We will start with the description of geometric points of "Hodgetype RapoportZink spaces" and the completed local rings thereof. Some applications to the study of Hodgetype Shimura varieties will be given.
13/05/15 4.00 AnneMaria ErnvallHytonen (Helsinki)
Title: On Baker type bounds and generalised transcendence measure
(joint work with K. Leppälä and Tapani Matalaaho)
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 2dimensional 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
deltapotential, on arithmetic 2Dtori. 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
RudnickUeberschaer, 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 Lfunctions. This is joint work with Byungchul Cha and Florent Jouve.
14/01/15 Lucio Guerberoff (UCL)
Title: Shimura varieties and complex conjugation.
Abstract: Work of Shimura, Langlands, MilneShih, 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 Lseries does not vanish at s = 1, and for which the full BirchSwinnertonDyer 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 padic 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 nonarchimedean 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 byproduct, 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: Lfunctions as distributions.
Abstract: In 1989, Selberg defined what came to be known as the "Selberg class" of Lfunctions, 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: padic local Langlands in weight 1.
Abstract: I will explain how one can recover the classical Langlands correspondence for GL(2,Qp) from the padic 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 padic nearby cycles.
Abstract: I will describe how the theory of (phiGamma) modules allows to relate padic 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 Mumfordtype 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 quasisplit group and a generic representation, they have been defined by the so called LanglandsShahidi method. One of the main ingredients in the LanglandsShahidi 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 nonarchimedean local field.
11/3/15 Andreas Langer (Exeter)
Title: GrothendieckMessing deformation theory for varieties of K3type.
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 K3type X (for example the Hilbert schemes of zerodimensional subschemes of a K3surface) is equipped with the additional structure of a 2display. Then we extend the GrothendieckMessing lifting theory from pdivisible groups to such varieties: The deformations of X over a nilpotent pdthickening correspond uniquely to selfdual liftings of the associated Hodgefiltration. For the proof we give an algebraic definition of the BeauvilleBogomolovForm 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 2display endowed with its BeauvilleBogomolov form. This is joint work with Thomas Zink.
18/3/15 Simon Wadsley (Cambridge)
Title: Dmodules 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 Dmodules for rigid analytic spaces in order to better understand the locally analytic representation theory of padic 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 'coadmissible modules' for this sheaf of noncommutative rings. When the rigid analytic space is the flag variety of a split semisimple padic analytic group then there is an equivalence of categories between the category of "coadmissible Dmodules" and coadmissible modules (in the sense of SchneiderTeitelbaum) 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.
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 nonhyperelliptic 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 r1 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 pseudocoefficient 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 vectorvalued 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: LondonParis 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 357 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 Fplectic 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 ("Fplectic") structure. I will talk about a first step towards showing the existence of such "canonical Fplectic 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)
Title: padic Langlands functoriality.
Abstract: In this talk we will study an example of padic 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 padic version of this transfer. More precisely we will extend the classical transfer to padic families of automorphic forms as parametrized by eigenvarieties. We will prove the padic 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 \elladic representations of GL_n(F), for F a padic 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 HeathBrown (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.
23/04/14 Jack Lamplugh (Cambridge University)
Title: Class numbers in some noncyclotomic $\mathbb{Z}_p$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 latticepoint 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 3dimensional 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 "Betticohomological overconvergent $p$adic modular forms" on $G$ under a very mild hypothesis (namely that $G$ splits over $F_v$ for each $vp$). 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 curvedefined 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 nonCM 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: LondonParis number theory seminar: details are here.
11/06/14 Minhyong Kim (Oxford University)
Title: "Nonabelian reciprocity laws and Diophantine geometry"
18/06/14 Andrew Wiles (Oxford)
Title: On class groups of imaginary quadratic fields
25/06/14 Zeev Rudnick (TelAviv 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.
15/01/14 Alex Bartel (Warwick)
Title: The CohenLenstra heuristic revisited
22/01/14 James Newton (Cambridge)
Title: Localglobal 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 SwinnertonDyer conjecture for nonabelian 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: padic measures for Hermitian modular forms and the RankinSelberg 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 Lfunctions and their symmetry
26/03/14 Jenny Cooley (Warwick)
Title: MordellWeil point generation on cubic surfaces over finite fields
2/10/13 Toby Gee (Imperial)
Title: The padic 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: Lfunctions of curves.
23/10/13 Rebecca Bellovin (Imperial)
Title: padic 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 BrauerManin 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: Gvalued 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.
24/4/13 2:30: P. CassouNoguès (Bordeaux)
Title: On simple rational polynomials.
24/4/13 4:00 (same day): Ph. CassouNoguès (Bordeaux)
Title: Cohomological invariants of quadratic forms
1/5/13 Igor Wigman (KCL)
Title: Nodal length fluctuations for arithmetic random waves.
8/5/13 Vladimir Dokchiter (Cambridge)
Title: Galois equivariance of Lvalues and the BirchSwinnertonDyer 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 pmodular automorphic representations of GL(2,Q_p)
29/5/13 Dinesh Thakur (Arizona)
Title: Higher multiplicities in number theory
34/6/13: LondonParis Number Theory Seminar (in London). Speakers: F. Charles (Rennes), Y. Harpaz (Nijmegen), JL. ColliotThelene (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: nonsolvable base change, symmetric powers and tensor products.
19/6/13 Christopher Daw (UCL)
Title: Degrees of strongly special subvarieties and the AndréOort conjecture.
26/6/13 Ashay Burungale
Title: Prigidity and Iwasawa muinvariants.
16/01 Speaker: David Loeffler (Warwick)
Title: Euler systems for the RankinSelberg 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 padic local Langlands correspondence and LubinTate groups
06/02 Speaker: Judith Ludwig (Imperial)
Title: padic functoriality for inner forms of unitary groups
13/02 Speaker: Gebhard Boeckle (Heidelberg)
Title: Independence of elladic 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 padic fields
06/03 NO SEMINAR
13/03 Speaker: Massimo Bertolini (Milan)
Title: Kato's reciprocity law and a padic Beilinson formula
20/03 Speaker: Damiano Testa (Warwick)
Title: The Buechi K3 surface and its rational points
03/10 Speaker: Andrei Yafaev (UCL)
Title: Hyperbolic AxLindemann theorem in the cocompact case
Abstract: This is a joint work with E. Ullmo. The classical AxLindemann 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: Fermattype 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 nontrivial 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 BarsottiTate deformation rings for twodimensional 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 TaylorWilesKisin patching argument, which, when combined with a new, geometric perspective on the BreuilMezard conjecture, forges a tight link between the structure of cohomology (a global automorphic invariant) and local deformation rings (local Galoistheoretic invariants).
Monday October 22nd: LondonParis Number Theory Seminar. Speakers: Sarah Zerbes (UCL), David Vauclair (Caen), David Burns (Kings College London).
24/10 Speaker: Tom Fisher (Cambridge)
Title: Visibility of TateShafarevich groups in abelian surfaces and abelian threefolds
Abstract: Let E be an elliptic curve over the rationals. The TateShafarevich 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 BreuilKisin modules.
Abstract: We will present an approach to the construction of BreuilKisin modules using crystalline cohomology. Along the way, we will define a new PDbase for crystalline cohomology that improves in some respects upon the ring S used by Breuil and Faltings in their appraoch to integral padic 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 3manifolds 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 HolowinskySoundararajan (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 BrauerManin 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: MordellLang in positive characteristic
Abstract: I will talk about the MordellLang conjecture in positive characteristic and sketch a new algebrogeometric proof for the case that the subgroup involved is finitely generated.
05/12 Speaker: Bruno Angles (Caen)
Title: The HerbrandRibet theorem for function fields revisited
Abstract: We will present a new proof of Taelman's HerbrandRibet 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 DeligneMumford stacks.
Abstract: We will give an overview of etale homotopy theory a la ArtinMazur of DeligneMumford stacks and discuss several examples including moduli stacks of algebraic curves and principally polarised abelian varieties.
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 BuzzardDiamondJarvis conjecture for unitary groups"
16 May: Riccardo Brasca, Milan
"padic modular forms of nonintegral weight over Shimura curves"
23 May: Konstantin Ardakov, QMUL
"Quillen's Lemma for affinoid enveloping algebras"
30 May: LondonParis 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
"The padic Geometric Langlands Correspondence"
27 June: Christian Johansson, Imperial
"Classicality for overconvergent automorphic forms on some quaternionic Shimura varieties"
Jan 18: Florence Gillibert (Bordeaux and MPI Bonn)
Title: Rational points on AtkinLehner quotients of Shimura curves
Jan 25 Roger HeathBrown (Oxford)
Feb 1st: Jan Nekovar (Paris)
Title: The method of Bertolini and Darmon
Feb 8: Richard Hill (UCL)
Feb 15 PaulJames White (Oxford)
Feb 22nd bonus talk at 2:30: Go Yamashita (Toyota)
Feb 22nd Vytas Paskunas (Bielefeld)
Feb 29th Jonathan Pila (Oxford)
Title: Some unlikely intersections beyond
AndreOort.
March 7th Rene Pannekoek (Leiden)
Title: On the padic density of the rational points on K3 surfaces.
March 14th Xavier Caruso
Title: An algorithm to compute the semisimplification modulo p of a
semistable representation
March 21 Anish Ghosh (Norwich)
Title: Effective density of rational points on homogeneous varieties
Speaker: Kevin Buzzard (Imperial College)
Title: Reduction of crystalline representations
Abstract: 2dimensional crystalline Galois representations of the absolute Galois group of Q_p can be completely determined by linear algebra data. One can also classify 2dimensional 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 padic 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 padic 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.

DATE: 26/10/11
Speaker: Francois Loeser (Jussieu)
Title: Motivic height zeta functions and the Poisson formula
Abstract: Recently, ChambertLoir 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 ChambertLoir.

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 localglobal principles.

DATE: 09/11/11
Speaker: Toby Gee (Imperial College)
Title: padic Hodgetheoretic 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 padic 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 KummerVandiver 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 ladic 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 HarrisTaylor/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 HasseWeil zeta function in terms of automorphic Lfunctions in nonendoscopic cases, and reconstruct the ladic 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 RhamWitt complex by imposing a growth condition on the de RhamWitt complex of DeligneIllusie 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 MonskyWashnitzer cohomology (considered before inverting p) of a smooth kalgebra is of finite type modulo torsion. This is joint work with Thomas Zink.
SPEAKER: Alan Lauder (Oxford)
TITLE: "Computations with classical and overconvergent modular forms"
ABSTRACT: We present padic 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: "SatoTate 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, ShepherdBarron and Taylor on the conjecture of SatoTate 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.

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 Lseries 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: "Pseudoabelian 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 pseudoabelian 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 pseudoabelian variety by a smooth connected affine group, in a unique way.
We work out much of the structure of pseudoabelian varieties. These groups are closely related to unipotent groups in characteristic p and to pseudoreductive groups as studied by Tits and ConradGabberPrasad. Many properties of abelian varieties such as the MordellWeil theorem extend to pseudoabelian varieties.

DATE: 15/6/11
SPEAKER: Wojciech Gajda (Adam Mickiewicz University, Poznan)
TITLE: "Independence of \elladic representations over function fields and abelian varieties"
ABSTRACT: Let K be a finitely generated field extension of Q. We consider a family of \elladic representations (\ell varies through rational primes) of the absolute Galois group of K, attached to the \elladic 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 \elladic 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 HigherOrder StickelbergerType 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 Lfunctions. We describe evidence in support of this conjecture, including a full proof in several important cases.
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 SiegelEisenstein 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 2nd Vadim Schechtman (Toulouse)
March 9th Jean Gillibert (Bordeaux) "On TateShafarevich groups and class groups"
Abstract: We discuss here a link between ShafarevichTate 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 socalled 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)
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 padic analytic geometry. 

8 November 
The LondonParis Number Theory Seminar speakers: F.Loeser (ENS, Paris), J.Pila (Bristol), B.Zilber (Oxford). 

10 November  JeanLouis ColliotThelene (Orsay) Unramified cohomology in degree 3. 

17 November  Pierre Lochak (Jussieu) Topological methods in GrothendieckTeichmueller theory. 

24 November  Jon Pridham (Cambridge) 

1 December  Tomer Schlank (Jerusalem) 

8 December  Tony Scholl (Cambridge) Hypersurfaces and the Weil conjectures 

15 December  TBA 
12 May  Matthias Flach (Caltech) "Weiletale cohomology of regular arithmetic schemes" 

19 May  Jared Weinstein (UCLA) "Resolution of singularities on the LubinTate 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 onedimensional formal module. We'll review this result, and show how the question of constructing nonabelian extensions leads to the study of the LubinTate tower, which can be viewed as an infinitesimal version of the classical tower of modular curves $X(p^n)$. By results of HarrisTaylor and Boyer, the cohomology of the LubinTate tower encodes precise information about nonabelian extensions of the local field (namely, it realizes the local Langlands correspondence). The LubinTate 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 followup 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 LondonParis Number Theory Seminar speakers: F. Brown, M. Hindry, M. Kakde 

9 June  Peter SwinnertonDyer (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 twodimensional Artin representations and weight one forms, and prove new cases of the strong Artin conjecture for totally odd, twodimensional "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 PerrinRiou), and gives a bound on the BlochKato Selmer group in terms of an r x r determinant. I will give two fundamental applications of this refinement: The first with the (conjectural) RubinStark elements; and the second with PerrinRiou's (conjectural) $p$adic $L$functions. 
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 finitedimensional kalgebra. Let d be an integer. Denote by Gr(d,A) the Grassmannian of dsubspaces of A (viewed as a kvector space), and by GL_1(A) the algebraic kgroup 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 SeveriBrauer 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 SatoTate conjecture for Hilbert modular forms" Abstract: I will discuss the SatoTate conjecture for Hilbert modular forms, which I recently proved in collaboration with Thomas BarnetLamb and David Geraghty. 

10 February  Jonathan Pila (Bristol) "A modeltheoretic approach to problems of ManinMumfordAndreOorttype" Abstract: I will describe a result, joint with Alex Wilkie, about the distribution of rational points on certain nonalgebraic sets in real space. The natural setting is an 'ominimal structure over the real numbers', a notion from modeltheory. A surprising strategy, proposed by Umberto Zannier, uses this result to approach diophantine problems in the ManinMumfordAndreOort circle of conjectures. I will describe some implementations of this strategy, including an unconditional proof of the AndreOort 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 ladic 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) "Nonabelian 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. 
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 MordellWeil group $J(K)$. Chabauty is a practical method for explicitly computing $C(K)$ provided $r \leq g1$. In unpublished work, Wetherell suggested that Chabauty's method should still be applicable provided the weaker bound $r \leq d(g1)$ 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/OdaMatsumoto Conjecture" Abstract: In his "Esquisse d'un programme", Grothendieck suggested that one should be able to give a nontautological 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 OdaMatsumoto. 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 HeathBrown (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 LondonParis 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 Qvector 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 pparity 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 HasseWeil Lfunction of E/F at s=1 and the corank of the pSelmer group of E/F have the same parity (joint work with C. Wuthrich). 

2 December  Behrang Noohi (King's) "Galois cohomology of crossedmodules and cohomology of reductive groups" Abstract: A 2group (or a crossedmodule) is a categorified version of a group. Line bundles over a scheme, for instance, form the Picard 2group. Galois cohomology of 2groups 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 $\mathbb{F}_1$. 
8 April  Mehmet Haluk Sengun (DuisburgEssen) "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 (ParisJussieu) "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 nonArchimedean field. The second construction is a purely algebraic approach to the regular representations of $GL_N(\mathfrak{o})$, which is formally similar to the BushnellKutzko 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 Lfunctions 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 HalterKoch 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 SwinnertonDyer conjecture with Artin formalism. 

20 May  Andreas Langer (Exeter) "Torsion zero cycles and padic integration theory" Abstract: We study the Chowgroup of zerocycles on the selfproduct of a CMelliptic curve over the field of padic numbers and prove that its pprimary torsion subgroup is finite, provided that p is an ordinary good reduction prime and the padic Lfunction 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 padic integration theory and Besser's computation of syntomic regulators for K_2 of curves and K_1 of surfaces. 

27 May two talks room 2B08 
2:003:00, room 2B08 Nigel Boston (UCDublin, Wisconsin) "The fewest primes ramifying in a Gextension of Q" Abstract: If G is a finite group, what is the smallest number of primes ramifying in a Gextension of the rationals? We give evidence for a conjectural answer, together with a conjectural density for such ntuples. [Parts are joint work with EllenbergVenkatesh and Markin.] 3:304:30, room 2B08 JeanPierre Serre "Variation with p of the number of solutions mod p of a given family of equations" 

3 June  4 June 
The LondonParis Number Theory Seminar at King's College London, room 2B08 theme: padic modular forms speakers: Buzzard, Fargues, Loeffler, Colmez, Mokrane, Dimitrov, Panchishkin 

10 June cancelled 
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 padic curves and padic representations" Abstract: The classical NarasimhanSeshadri 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 padic setting. We will report on recent progress and the main open questions in this area. There is related work of Faltings on padic Higgs bundles. 
14 January  Victor Abrashkin (Durham) "padic semistable representations and generalization of the Shafarevich Conjecture" Abstract: Breuil's theory of semistable padic 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 padic 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 Qrational points on the hypersurface, provided that the dimension is sufficiently large. Thanks to work of Davenport, and more recently of HeathBrown, 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 Lfunctions" 

25 February TWO TALKS 
2:003:00 Hershy Kisilevsky (Concordia) "Critical values of derivatives of (twisted) elliptic Lfunctions" 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 nonzero, 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 $\ord_{s=1}L(E,s,\chi)>1.$ 4:005:00 Christian Wuthrich (Nottingham) "Selfpoints 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 selfpoint, 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 semistable), one can prove that the point is of infinite order in the MordellWeil 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 March TWO TALKS 
2:003:00 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, ColliotThelene, Ekhedal on Hilbert's irreducibility theorem and to some classical theorems of Grunwald and Neukirch. (This is a joint work with Nour Ghazi). 4:005:00 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 
1 October 14.0015.00 room 342 
Thomas Zink (Bielefeld) "pdivisible groups over regular local rings of mixed characteristics" 

8 October  Go Yamashita "Upper bounds for the dimensions of padic 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 Ktheory and Modular Symbols" 

29 October  Tejaswi Navilarekallu (Vrije Universiteit Amsterdam) "Equivariant padic Lvalues" 

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 LondonParis 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" 
23 April  Henri Johnston (Oxford) "Nonexistence 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 padic automorphic forms" 

7 May  8 May 
The LondonParis Number Theory Seminar  
13 May Tuesday 3:304:30 room 436 
Cristian Popescu (UCSD) "On the CoatesSinnott 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 padic valuation of the values of normalized Hecke eigenforms attached to nonisotrivial 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 MordellWeil 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 MordellWeil 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 noncommutative Main Conjecture for totally real number fields" 

25 June  Jonathan Pila (Bristol) "Rational points of definable sets and the ManinMumford conjecture" Abstract: I will discuss problems and results concerning the distribution of rational points on certain nonalgebraic sets. More specifically, definable sets in ominimal 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 semialgebraic 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 ManinMumford conjecture by combining these ideas with a result of Masser. 
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 twodimensional 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 equidistributed, 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 multivariable polynomials and show that such polynomials must have low rank. We apply this result to the study of socalled Gowers norms. Let P : F^n > F be any function. It is wellknown 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 talks 4 pm6 pm 
4 pm5 pm:
Takako Fukaya (Keio and Cambridge) "Root numbers, Selmer groups, and noncommutative Iwasawa theory" 5 pm6 pm: Kazuya Kato (Kyoto and Cambridge) "Classifying spaces of mixed Hodge (resp. padic Hodge) structures" 

13 February  Nick ShepherdBarron (Cambridge) "Thomae's formula for nonhyperelliptic curves" Abstract: In 1857 Thomae gave formulae for the thetaconstants 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 nonhyperelliptic curves. 

20 February  Fabien Trihan (Nottingham) "Crystalline representations and Fcrystals" 

27 February  Andreas Doering (Imperial) "Topos theory in the foundations of physics" 

5 March  Burt Totaro (Cambridge) "Moving codimensionone subvarieties over finite fields" 

12 March  Tamas Hausel (Oxford) "Arithmetic harmonic analysis on character and quiver varieties" 
3 October  David Loeffler (ICL) "Overconvergent padic automorphic forms and eigenvarieties for compact reductive groups" Abstract: I shall describe a construction of an eigenvariety parametrising padic 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 "semiclassical" 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 groupring which annihilates the imaginary part of the class group of an abelian field. In the 1980s Tate and Brumer proposed a generalisation (the "BrumerStark 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 padic 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 dth symmetric power of a curve provided the rank of the MordellWeil group of the Jacobian is at most gd (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 HodgeStructures 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 RapoportZink 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 ColmezFontaine 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 LondonParis 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 lmodular representations. 

21 November 4 pm5:30 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 CastanoBernard (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 Lfunction has a simple zero at s = 1. In particular, there is a nonconstant 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 Qrational. Moreover, it is wellknown 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 orless likelythe TateShafarevich group of E is nontrivial. 

5 December  Laurent Fargues (Université ParisSud, Orsay) "Ramification of LubinTate groups and the BruhatTits building" Abstract: One of the purposes of this talk is to give a description of the isomorphism between the padic LubinTate and Drinfeld towers at the level of their skeletons. For the Drinfeld space, its skeleton is the BruhatTits building of the linear group. For example, we can describe explicitly the pullback of this simplicial structure on the open padic ball associated to the LubinTate space. We also study in detail the different ramification filtrations (upper and lower) associated to LubinTate 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" 
25 April  Shaun Stevens (UEA) "Supercuspidal representations of padic classical groups" 

2 May 
The LondonParis 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 wellknown; in higher dimensions, however, the situation is more complicated, and few explicit nontrivial examples of isogenies are known. We will describe some interesting examples of explicit isogenies of Jacobians of lowgenus curves, discuss some of the computational issues, and give some applications in modern cryptography. 

16 May  Matthias Strauch "Potentially crystalline representations and associated padic 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 twodimensional 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" 
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 halfplane 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 \ZGmodule. 

31 January  David Burns (King's) "Iwasawa theory of elliptic curves in padic 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 Frobsemisimple WeilDeligne representation. Classical LocalGlobal compatibility ensures that these agree under the (correctly normalised) local langlands correspondence. I will discuss ways of attaching such objects to overconvergent padic eigenforms across the eigencurve and to what extent localglobal compatibility remains valid. 

14 February  Tom Fisher (Cambridge) "Finding rational points on elliptic curves using 6descent and 12descent" Abstract: Descent on an elliptic curve E is used to obtain partial information about both the group of rational points (the MordellWeil group) and the failure of the Hasse principle for certain twists of E (the TateShafarevich group). The Selmer group elements computed may be represented as ncoverings of E. Traditionally one takes n to be a prime power. Breaking with this tradition, I explain how to combine the data of an mcovering and an ncovering, for m and n coprime, to obtain an mncovering. This technique improves the search for rational points on E. In particular using 6descent and 12descent, I was recently able to find all the "missing" generators for the elliptic curves of analytic rank 2 in the SteinWatkins database. 

21 February  Daniel Delbourgo (Nottingham) "Nonabelian congruences between Lvalues 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 padic 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 padic 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 BlochKato conjecture. We further show how to obtain instances of the FontaineMazur conjecture for imaginary quadratic fields in the residually reducible case. The latter is joint work in progress with Kris Klosin. 
4 October  Andrei Yafaev (UCL) "On the triviality of rational points on certain AtkinLehner 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 AtkinLehner 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 pisogeny and semistable at primes above p. Then one can determine the root number and the parity of the pSelmer 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 padic families" 

8 November  Sarah Zerbes (Imperial) "Higherdimensional logarithmic derivatives" Abstract: In my talk, I will explain how to construct logarithmic derivative maps for ndimensional local fields of mixed characteristic (0,p). The main ingredients for this construction are higherdimensional 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
LondresParis à l'Institut Henri Poincaré
programme Orateurs: Fred Diamond, Toby Gee, Florian Herzig.  
15 November  Andres Helfgott (Bristol) "How small must illdistributed 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 socalled classinvariant 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 ``subgroupfree" approach to Canonical Subgroups" Abstract: I will discuss joint work with E. Goren in which we present a ``subgroupfree'' 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. 
3 May 
Alan Lauder (Oxford)  
10 May 
Carlos CastanoBernard (Cambridge) 

17 May 
Ben Smith (Royal Holloway) 

24 May 
Rob de Jeu (Durham) 

31 May 
Herbert Gangl (Durham) 

7 June 
Guy Henniart (Paris) 

14 June 
Otmar Venjakob (Bonn) 

21 June 2.30pm 
Minhyong Kim (Purdue) 

21 June 4.00pm 
Martin Taylor (Manchester) 

26 July 2.00pm 
Doug Ulmer (U. Arizona, Tucson) 
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 AndreOort conjecture" Abstract: This is a joint work with Bruno Klingler. We explain a proof of the AndreOort conjecture under the assumption of the generalised Riemann Hypothesis.  
1 February  David Burns (KCl) "Algebraic padic Lfunctions in noncommutative Iwasawa theory" Abstract: There have been several important developments in noncommutative Iwasawa theory over the last few years. We discuss a natural construction of algebraic padic Lfunctions in this setting and discuss some interesting consequences for the main conjectures of noncommutative Iwasawa theory formulated by Coates, Fukaya, Kato, Sujatha and Venjakob and by Fukaya and Kato.  
8 February  Daniel Caro (Durham) "Towards a good padic cohomology" Abstract: First, we will trace the history of the search for a good padic cohomology over schemes in characteristic p. We arrive at the construction of Berthelot's arithmetical Dmodules. We will explain why these objects now represent the only possibility of obtaining a good padic cohomology. An important result which inspires trust in this theory is the following: for every overholonomic Fcomplex E of arithmetical Dmodules 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 Fisocrystals). 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 2dimensional 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, Zextensions of C*, the exponential function, and a homotopytheory 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 nth 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 Zetafunctions'. 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. 
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 wellknown 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 DeligneMumford) 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 Lfunctions 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 BrumerStark 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 padic 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 psemilocal 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 nonsingular 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 HeathBrown, we discuss the final resolution of this conjecture. 
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 SwinnertonDyer (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 Lfunction) 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 halfplane Re(s)>constant. If one tries the same trick replacing the complex numbers with the padics, 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 Lfunctions 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 padic Lfunctions of elliptic curves.  
25 May  Chris Skinner (Michigan) "Lvalues, 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 1980s 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 AndreOort conjecture"  
15 June  Teruyoshi Yoshida (Harvard and Nottingham) Compatibility of local and global Langlands correspondences" (with R.Taylor) Abstract: The work of HarrisTaylor, which proved the local Langlands correspondence for GLn, included the construction of ladic 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 TateShafarevich group" 

29 June  Bjorn Poonen (Berkeley and Cambridge) "Multiples of subvarieties in algebraic groups over finite fields" 
12 January  Michael Harris (Universite Paris VII) "Deformations of automorphic Galois representations" Abstract: The ladic cohomology of Shimura varieties attached to certain unitary groups provide compatible systems of ndimensional ladic representations of the absolute Galois group of a CM field. These representations are necessarily polarized (selfdual, more or less) and their HodgeTate 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 BrauerManin Obstruction on Curves" Abstract: When a variety violates the Hasse principle, this can be due to an obstruction known as the BrauerManin 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 BrauerManin 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 ShafarevichTate 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 ColemanVoloch. 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 ColemanVoloch, 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 nonsimply connected homotopy theory (and thereby also in combinatorial group theory) to decide whether, over a given fundamental group G, every algebraic 2complex 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 noncancellation phenomena for modules over orders in quaternion algebras over number fields, we describe families of algebraic 2complexes 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 Lfunction L(E/K,s), where K varies over different number fields. When the sign is 1, the Lfunction (assuming it is exists and is analytic) has a zero at s=1, and the BirchSwinnertonDyer 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 (noncube) m>1. I will discuss this, and similar phenomena.  
16 February  Andreas Langer (Exeter) "GaussManin connection via Wittdifferentials" Abstract: For a scheme X that is smooth over a padic base we show an equivalence of categories between the category of locally free crystals and the category of Wittconnections on X. The proof uses the relative de RhamWitt complex and generalizes a recent result of Bloch. As an application we realize the GaussManin connection in the de RhamWitt complex.  
23 February  KyuHwan Lee (U. Toronto) "Spherical Hecke algebras of GL_n over 2dimensional 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 2dimensional local field case. More precisely, we will construct spherical Hecke algebras of GL_n over 2dimensional local fields and prove the Satake somorphism for the algebras. We will use Fesenko's measure to define the Satake isomorphism. A connection to KacMoody 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 Lfactors are automorphic, using LaplaceCarleman 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. 
27 October  No seminar  
3 November  Pierre
Parent (Bordeaux) "On the triviality of X_{0}^{+} (p^{r}) (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 )> GL_{2} (F_{p}) induced by the Galois action on the group of ptorsion points of E. A theorem of Serre, published in 1972, asserts that there exists an integer B_{E} such that the above representation is surjective if p is larger than B_{E}. Serre then asked the following question: can B_{E} 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 X_{0} (p), X_{}_{split}(p), X_{}_{nonsplit}(p) and X_{A4}(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 (socalled exceptional) case of X_{A4}(p) was ruled out by Serre. The fact that X_{0} (p)(Q ) is made of only cusps for p>163 is a wellknown theorem of Mazur. In this talk we will discuss the case of X_{}_{split} (p)(Q). Slightly more generally (because one has a Qisomorphism between X_{}_{split} (p) and X_{0}^{+} (p^{2} )), we will in fact give a criterion for the triviality of X_{0}^{+} (p^{r} ) (Q) (with r>1), and show it is verified by a positive density of primes (satisfying explicit congruences).  
10 November  Nick
ShepherdBaron (Cambridge) "Perfect forms and moduli of abelian varieties" Perfect quadratic forms lead to a compactification of A_{g} 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 FontaineMazur 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 Q_{p}((t)), C((s))((t)). Should be of interest to geometers also this can be respelled as theorems about the moduli of Gbundles on an algebraic surface "Donaldson theory".  
1 December  Shaun Stevens (UEA) "Supercuspidal representations of padic classical groups" The Local Langlands Correspondence relates the representations of the WeilDeligne group of a locally compact nonarchimedean local field to the irreducible smooth representations of general linear groups. Mostly conjecturally, there are also such correspondences for other padic groups. In this talk I will try to describe what this means, what's known and also some explicit constructions of representations for padic 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. 
28 April  Dan Snaith (Imperial) "Overconvergent Siegel modular forms"  
5 May  Luis
Dieulefait (Barcelona) "Existence of compatible families of Galois representations and the FontaineMazur conjecture for elliptic curves"  
12 May  Bruno Kahn (Paris
7) "Birational motives"  
19 May  Fre
Vercauteren (Bristol) "Zeta functions: the padic approach"  
26 May  Sarah
Zerbes (Cambridge) "Selmer groups over padic 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 mid70s, 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) "Nonabelian LubinTate theory and DeligneLusztig theory"  
4 August  Chandrashekhar Khare
(Utah/Tata) "Transcendental Galois representations" 
14 January  Kevin Buzzard
(Imperial) "The 2adic 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 mid1990s 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 nontrivial though very small fundamental group. For rational surfaces it is conjectured by ColliotTh 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 Qrational points. The proof uses descent to a torsor on this surface; the structure group of this torsor is a form of a (nonabelian) 1dimensional orthogonal group.  
28 January  Roger
HeathBrown (Oxford) "Cayley's cubic surface" Roughly how many nontrivial primitive integer solutions does the equation 1/X_{0}+1/X_{1}+1/X_{2}+1/X_{3}=0have, in a large cube max X_{i} < 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 Lfunctions 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 Lfunctions, and Gross, who gave congruences for the value of the Lfunction. These conjectures admit a natural refinement, due to Burns, which gives Grossstyle congruences in the situation of Rubin's conjecture. The formulation of the refinement is inspired by the Equivariant Tamagawa Conjecture, and this overriding 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 rayclass 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) "padic 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 nth 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 padic Tate twists on regular arithmetic schemes (=schemes which are regular flat of finite type over Spec Z), and arithmetic duality theorems for padic Tate twists, which generalizes the classical ArtinVerdier 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 Lfunctions" Secondorder modular forms are functions that have recently appeared in several contexts: Eisenstein series formed with modular symbols, converse theorems of Lfunctions, 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 Lfunctions.  
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. 
1/10/03 David Solomon, KCL `Twisted ZetaFunctions, 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 padic modular forms"  12/11/03 Neil Dummigan, Sheffield, `Critical values of tensorproduct Lfunctions'  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: `HopfGalois strucutres of field extensions'  10/12/03 Prof. Igor Shparlinski, Macquarrie U. `Euler Function: Smooth and Square' 
Further details will appear when available.
This term it was held in the maths department of Imperial College and was
organised by
Dr Kevin Buzzard (buzzard@ic.ac.uk).
 30 April  TWO TALKS 14:15 Elmar GrosseKloenne (Muenster) "On twisted unit root Lfunctions ^^^^^ of families of varieties over finite fields" 15:45 Steve Gelbart (Weizmann) "On lower bounds for automorphic ^^^^^ Lfunctions".  7 May Allan Lauder (Oxford) "Deformation theory and the computation of zeta functions"  14 May Vic Snaith (Southampton) "On the KummerVandiver conjecture"  ***Tuesday 20th May*** at 1600 Victor Rotger (Barcelona)"Diophantine properties of fake elliptic curves and their moduli spaces"  21 MayEd Nevens (Imperial)TBA (something about moduli space of abelian varieties and/or canonical subgroups, perhaps)  28 MayNeil Strickland (Sheffield)"Elliptic cohomology" 
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).
 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 2adic 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 padic 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 conjectureswith a DIFFERENCE"  5/2 A. Hayward (Kings) "A conjectural classnumber formula for higher derivatives of abelian Lfunctions"  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) "FourierJacobi 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 1motives"  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" 
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 DOUBLEHEADER 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 padic field with good reduction and let Y be its special fiber. We consider the structure of the Chow group CH0(X) of zerocycles 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 pprimary torsion. On the contrary, we shall show that the pprimary 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 Lfunctions 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 Cstructures on Rn have a Hermitian form compatible with the Euclidean structure on Rn and for which L becomes an Omodule for some quadratic order O. In some cases we determine explicitly the Omodule structure of L.  4/12 David Burns, King's College London Title: `Nearly perfect complexes and Weiletale 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 (padic) 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 K2n1(k) for n>=2. The Kgroups 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 padic regulator) in this context, involving polylogarithms.
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 GL_{2}(F_{7}) 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 2adic 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"
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 epsilonfactors 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 nonAbelian 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_pextensions"
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 SwinnertonDyer (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 2covering 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 2division points defined over k, I shall show that the Hasse Principle holds for V provided that (i) the TateSafarevi\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 BrauerManin 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 padic Lfunctions" Oct 31 Jonathan Dee (Imperial) "PhiGammamodules and families of padic Galois representations" Nov 7 JeanLouis ColliotTh e (Orsay) "Linear algebraic groups over twodimensional 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 padic 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 ColemanMazur 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 Gvariety 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 is known about this problem, both for the classical groups and the exceptional groups such as E_8. Dec 5 Otmar Venjakob (Cambridge) "Iwasawa theory of padic Lie extensions" Abstract: The most prominent example of a (nonabelian) padic Lie extension K of a number field k arises maybe by adjoining to k the ppower 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 pseudonull 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 pseudoisomorphism, which was proven by Coates, Schneider and Sujatha recently. Dec 12 No Talk
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) 
(Final Seminar) 
Jun 26  Frank Calegari
Fontaine proved in 1985 that there do not exist any Abelian varieties over Z. Jacobians of the modular curves X_0(p^n) (nonzero 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 BrumerKramer 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
During the past few years, Barry Mazur, myself, and others have studied visible subgroups of ShafarevichTate groups of abelian varieties. Recently, I've been studying visibility of MordellWeil groups of abelian varieties. In this talk, I will very briefly review some results about visibility of ShafarevichTate groups, then discuss some of what I've been able to prove about their counterparts in the context of visibility of MordellWeil groups. In particular, I will show that MordellWeil groups of elliptic curves over Q are visible in modular abelian varieties. 
Jun 15  Helena Verrill

Jun 13  Romyar Sharifi 
Abstract: We give an explicit description of generators of the ith unit groups of K = Q_{p}zeta_{pn} as Galois submodules of the multiplicative group of K. We can use this to determine the ramification groups of degree p^{n} Kummer extensions of K which are Galois over Q_{p}. 
Jun 6  Richard Hill (University College, London) 

May 30  Prof. V. Nikulin (Liverpool) 

May 23  Daniel Delbourgo (Nottingham) 

My 16  Rob de Jeu (Durham) 

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) 

Jan 24  Al Weiss (U Alberta)

Jan 31  Andreas Langer (Bielefeld)

Feb 7  Colin Bushnell (Kings)

Feb 14  Alexei Skorobogatov (Imperial)

Feb 21  Christophe Cornut (Strasbourg)

Feb 28  Keith Ball (UCL)

Mar 7  Werner Hoffman (Humboldt U)

Mar 14  Anupam Saikia (Cambridge)

Mar 21  Victor Flynn (Liverpool)

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. 
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)

Oct 18  Alexei Skorobogatov (Imperial)

Oct 25  David Solomon (KCL)

Nov 1  Kevin Buzzard (IC)

Nov 8  Jan Nekovar (Cambridge)

Nov 15  Tom Fisher (Cambridge)

Nov 22  Roger HeathBrown (Oxford)

Nov 29  JeanRobert Belliard (Nottingham)

Dec 6  Richard Hill (UCL)

May 24  Richard Hill (UCL)

May 31  Frazer Jarvis (University of Sheffield)

Jun 7  Alexei Skorobogatov (ICL)

Jun 14  Anton Deitmar (Exeter)

Jun 21  No seminar scheduled 
Jun 27  Cornelius Greither (U. der Bundeswehr, Munich)

Jan 26  Neil Dummigan (Oxford)

Feb 2  Susan Howson (Nottingham)

Feb 9  David Burns (KCL)

Feb 16  Victor Abrashkin

Feb 23  David Burns (KCL)

Mar 1  Richard Hill

Mar 8  Robert Vaughan

Mar 15  David Solomon

Mar 22  David Solomon

From the 4th of November, the seminars moved to Imperial College and were organised by Dr Kevin Buzzard.
CambridgeOxfordWarwick (COW) algebraic geometry seminar, room 642 IC  
2.00  Nick ShepherdBarron (Cambridge CMS) 
 
3.15  Paul Seidel 

Abstract:
In 1989 Glenn Stevens showed how the `periods' of
Eisenstein series could be used to define certain families of 1cocycles
on GL_{2}(Q) whose values can be expressed in terms of Dedekind
Sums (for example, those appearing in the transformation formula for
Dedekind's etafunction). 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 zetafunctions of real quadratic fields at nonpositive integers.
In 1992, Robert Sczech constructed 1cocycles on GL_{2}(Q) by a
different method, using only real analysis. He too showed how they related
to Dedekind Sums and partial zetavalues. (Indeed they are very closely
related to Stevens'). In 1993 Sczech extended his construction to
GL_{n}(Q), producing (n1)cocycles that are similarly `universal'
for the evaluation of partial zetavalues over totally real fields of
degree n.
In this talk, I shall present a third construction of cocycles on
GL_{n}(Q) for n=2 and 3, which was inspired by Shintani's formulae
for partial zetavalues. 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 zetavalues, padic interpolation, and the `challenge of higher
dimensions'.
Abstract:
Let S be an algebraic group over a field k, and X be a
kvariety. 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
kpoint, then this class is neutral. In some arithmetically meaningful
cases, e.g. X a principal homogeneous space of a semisimple 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).
Abstract:
The talk will be about a nonclassical generalisation of
the CasselsTate pairing. It is defined, roughly speaking, on the
nongeneric part of the dual of the Selmer group. The existence of the
pairing has nontrivial consequences; for example, one can deduce results
about the parity of the (co)rank of the Selmer group.
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.