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 Chris Blake (Cambridge)
Title: The F-plectic Taniyama group

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.

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 Lucio Guerberoff
Title:

26/11/14 Judith Ludwig (Imperial)
Title: p-adic Langlands functoriality.

3/12/14
Title:

10/12/14
Title:

17/12/14
Title:

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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