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 2018), the seminar will be hosted by Imperial College, and will be held on Wednesdays from 4pm to 5pm in room 340 (note room change) in the Huxley Building. The seminar is organized by Ana Caraiani, Carl Wang-Erickson and Chris Williams, and runs from Wed 3rd October to Wed 12th December (inclusive).

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

3 Oct 2018 - Adam Harper (Warwick)
Title: Low moments of character sums
Abstract: Sums of Dirichlet characters $\sum_{n \leq x} \chi(n)$ (where $\chi$ is a character modulo some prime $r$, say) are one of the best studied objects in analytic number theory. Their size is the subject of numerous results and conjectures, such as the Pólya-Vinogradov inequality and the Burgess bound. One way to get information about this is to study the power moments $\frac{1}{r-1} \sum_{\chi \text{mod } r} |\sum_{n \leq x} \chi(n)|^{2q}$, which turns out to be quite a subtle question that connects with issues in probability and physics. In this talk I will describe an upper bound for these moments when $0 \leq q \leq 1$. I will focus mainly on the number theoretic issues arising.

10 Oct 2018 - Sarah Zerbes (UCL)
Title: Euler systems for Siegel modular forms
Abstract: Euler systems are compatible families of cohomology classes attached to global Galois representations, which play a fundamental role in relating values of L-functions to arithmetic. I will sketch the construction of an Euler system for the spin representation attached to genus 2 Siegel modular forms. The construction reveals a surprising link to branching laws in local representation theory and the Gan-Gross-Prasad conjecture. This is joint work with D. Loeffler and C. Skinner.

17 Oct 2018 - Netan Dogra (Oxford)
(This week the seminar will be in Huxley room 213)
Title: Serre's uniformity question and the Chabauty-Kim method for modular curves
Abstract: Serre's uniformity question asks which Galois representations can arise from the $p$-torsion of an elliptic curve over $\mathbf{Q}$. Equivalently, it can be viewed as a question about rational points on certain modular curves. In this talk, I will explain what is known about the problem, and describe some recent joint work with Samuel Le Fourn and Samir Siksek on understanding these rational points via the Chabauty-Kim method.

24 Oct 2018 - Andrea Dotto (Imperial)
Title: Diagrams in the mod $p$ cohomology of Shimura curves.
Abstract: In search of a local mod $p$ Langlands correspondence, one can study globally defined representations that should correspond to a given local Galois representation: for example, those arising from completed cohomology or from spaces of algebraic modular forms. Then there's the issue of proving that these representations are independent of the global context. I will present some recent progress on this problem for mod $p$ representations of the group $\mathrm{GL}(2)$ over finite unramified extensions of $\mathbf{Q}_p$, answering a question of Breuil about an analogue of Colmez's functor. This is joint work with Daniel Le.

31 Oct 2018 - Vytas Paskunas (Essen)
Title: On some consequences of a theorem of J. Ludwig
Abstract: We prove some qualitative results about the $p$-adic Jacquet--Langlands correspondence defined by Scholze, in the $\mathrm{GL}(2,\mathbf{Q}_p)$, residually reducible case, by using a vanishing theorem proved by Judith Ludwig. In particular, we show that in the cases under consideration the $p$-adic Jacquet--Langlands correspondence can also deal with principal series representations in a non-trivial way, unlike its classical counterpart. The paper is available at http://arxiv.org/abs/1804.07567.

7 Nov 2018 - Preston Wake (IAS)
Title: Variation of Iwasawa invariants in residually reducible Hida families
Abstract: We'll discuss a work in progress describing properties of $p$-adic $L$-functions of a modular form whose Galois representation is residually reducible. As an application, we prove cases of a conjecture of Greenberg about $\mu$-invariants of Selmer groups. This is joint work with Rob Pollack.

14 Nov 2018 - Victor Rotger (Barcelona)
Title: Venkatesh's conjecture for modular forms of weight one
Abstract: Akshay Venkatesh and his coauthors (Galatius, Harris, Prasanna) have recently introduced a derived Hecke algebra and a derived Galois deformation ring acting on the homology of an arithmetic group, say with $p$-adic coefficients. These actions account for the presence of the same system of eigenvalues simultaneously in various degrees. They have also formulated a conjecture describing a finer action of a motivic group which should preserve the rational structure $H^i(\Gamma,\mathbf{Q})$. In this lecture we focus in the setting of classical modular forms of weight one, where the same systems of eigenvalues appear both in degree 0 and 1 of coherent cohomology of a modular curve, and the motivic group referred to above is generated by a Stark unit. In joint work with Darmon, Harris and Venkatesh, we exploit the Theta correspondence and higher Eisenstein elements to prove the conjecture for dihedral forms.

21 Nov 2018 - Jack Shotton (Durham)
Title: Shimura curves and Ihara's lemma
Ihara's lemma is a statement about the structure of the mod l cohomology of modular curves that was the key ingredient in Ribet's results on level raising. I will motivate and explain its statement, and then describe joint work with Jeffrey Manning on its extension to Shimura curves.

28 Nov 2018 - Yichao Tian (Strasbourg)
Title: Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives
Abstract: In my talk, I will report on my ongoing collaborating project together with Yifeng Liu, Liang Xiao, Wei Zhang, and Xinwen Zhu, which concerns the rank 0 case of the Beilinson-Bloch-Kato conjecture on the relation between L-functions and Selmer groups for certain Rankin-Selberg motives for GL(n) x GL(n+1). I will state the main results with some examples coming from elliptic curves, sketch the strategy of the proof. If time allows, I will also explain some key geometric ingredients in the proof, namely the semi-stable reduction of unitary Shimura varieties of type U(1,n) at non-quasi-split places.

5 Dec 2018 - Lucia Mocz (Bonn)

12 Dec 2018 - Jessica Fintzen (Cambridge)