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 2014), 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 Kevin Buzzard, and starts on 8th October (Imperial's term only starts on the 6th).

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

## The talks.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

26/11/14 Judith Ludwig (Imperial)

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

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

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

10/12/14 Roger Heath-Brown (Oxford)
Title: TBA

[no seminar 17/12/14]

The seminar is preceded by the Study Group, which this term will be on Vincent Lafforgue's recent work on the Langlands correspondence for function fields.

A list of previous seminar talks is here.

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This page is maintained, poorly, by Kevin Buzzard. If anyone thinks they can do better they're welcome to offer.