London Number Theory Seminar

The London Number Theory Seminar is held weekly, on Wednesdays, during term time. The location of the seminar cycles between KCL, Imperial College and UCL.

This term (Autumn 2016), the seminar will be hosted by Imperial College, and will be held on Wednesdays from 4pm to 5pm in room 140 in the Huxley Building. The seminar is organized by Stephane Bijakowski, Olivier Taibi and Rebecca Bellovin, and runs from 5th October to 14th December (inclusive).

The talks will be preceded by tea/coffee in Imperial's common room (room 549 Huxley) from around 3:30.

The speakers so far are below.

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

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

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

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

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

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

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

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

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

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

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

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

The seminar will be preceded by the Study Groups, and this term there may well be three of them, one running 1200-1330 and two running 1400-1530 concurrently. More details later.

A list of previous seminar talks is here.

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