For future Number Theory seminars see:

https://researchseminars.org/seminar/SheffieldNumberTheory


Past events:

Sheffield Number Theory Day, 30 June 2022


  • 11-12 Hanneke Wiersema (Cambridge)

  • 12-13 Lunch

  • 13-14 Tzu-Jan Li (Paris)

  • 14-14:30 Coffee break

  • 14:30-15:30 Chris Williams (Warwick)


All talks will be in Hicks building J-11.


Abstracts:


Hanneke Wiersema, Modularity in the partial weight one case

The strong form of Serre's conjecture states that a two-dimensional mod p representation of the absolute Galois group of Q arises from a modular form of a specific

weight, level and character. Serre considered modular forms of weight at least 2, but in 1992 Edixhoven refined this conjecture to include weight one modular forms. In this talk we explore analogues of Edixhoven's refinement for Galois representations of totally real fields, extending recent work of Diamond–Sasaki. In particular, we show how modularity of partial weight one Hilbert modular forms can be related to modularity of Hilbert modular forms with regular weights, and vice versa.


Tzu-Jan Li, On endomorphism algebras of Gelfand--Graev representations

Helm and Moss have recently studied a problem on "local Langlands correspondence in families" for the p-adic general linear groups, through which they have also obtained an invariant-theoretical description of integral endomorphism algebras of Gelfand--Graev representations of finite general linear groups. In this talk, we shall generalise Helm--Moss's result on endomorphism algebras of Gelfand--Graev representations to the case of any reductive groups having connected center. Instead of using Helm--Moss's p-adic approach, we will use the Brauer theory of modular representations to relate the endomorphism algebra in question with the desired invariant-theoretical description. This talk is mainly based on the work [1] in collaboration with Jack Shotton.

Reference:

[1] T.-J. Li and J. Shotton, On endomorphism algebras of Gelfand-Graev representations II, preprint (arXiv:2205.05601)


Chris Williams, p-adic L-functions for GL(3)

Let \pi be a p-ordinary cohomological cuspidal automorphic representation of GL(n,A_Q). A conjecture of Coates--Perrin-Riou predicts that the (twisted) critical values of its L-function L(\pi x\chi,s), for Dirichlet characters \chi of p-power conductor, satisfy systematic congruence properties modulo powers of p, captured in the existence of a p-adic L-function. For n = 1,2 this conjecture has been known for decades, but for n > 2 it is known only in special cases, e.g. symmetric squares of modular forms; and in all previously known cases, \pi is a functorial transfer via a proper subgroup of GL(n). In this talk, I will explain what a p-adic L-function is, state the conjecture more precisely, and then describe recent joint work with David Loeffler, in which we prove this conjecture for n=3 (without any transfer or self-duality assumptions).