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Number Theory Seminar

Unless otherwise specified, the seminars are held on Mondays at 15:00 in Room B3.03 – Mathematics Institute

2019-20 Term 1

Organiser: Martin Orr

7 October

Daniel Gulotta (University of Oxford)

Vanishing theorems for Shimura varieties at unipotent level

We prove a vanishing result for the compactly supported cohomology of certain infinite level Shimura varieties. More specifically, if X_{K_p K^p} is a Shimura variety of Hodge type for a group G that becomes split over Q_p, and K_p is a unipotent subgroup of G(Q_p), then the compactly supported p-adic etale cohomology of X_{K_p K^p} vanishes above the middle degree.
We will also give an application to eliminating the nilpotent ideal in the construction of certain Galois representations.
This talk is based on joint work with Ana Caraiani and Christian Johansson and on joint work with Ana Caraiani, Chi-Yun Hsu, Christian Johansson, Lucia Mocz, Emanuel Reinecke, and Sheng-Chi Shih.

14 October

Sam Chow (University of Warwick)

Rado's criterion over squares and higher powers

Given a finite colouring of the integers, is there a monochromatic Pythagorean triple? With Sofia Lindqvist and Sean Prendiville, we provide an affirmative answer in the analogous setting of generalised Pythagorean equations in five or more variables. Moreover, we show that a diagonal equation in sufficiently many variables has this property if and only if some non-empty subset of the coefficients sums to zero, which is a higher-degree version of Rado's characterisation of the linear case.
21 October Christopher Daw (University of Reading)
28 October Sarah Peluse (University of Oxford)
4 November Giada Grossi (UCL)
11 November

Chris Williams (University of Warwick)

p-adic L-functions for symplectic representations of GL(2n)

Let F be a totally real field, and let pi be an automorphic representation of GL(2n)/F that admits a Shalika model (that is, it is a transfer from GSpin(2n+1)). When pi is ordinary at p, recent independent work of Gehrman and Dimitrov--Januszewski--Raghuram gives a p-adic L-function attached to pi, that is, a p-adic measure interpolating its classical critical L-values. I will report on ongoing joint work with Daniel Barrera and Mladen Dimitrov where we generalise this to the non-ordinary case using overconvergent cohomology. Rather than standard overconvergent cohomology, defined with respect to the maximal torus in GL(2n), our results use a more flexible definition defined with respect to the subgroup GL(n) x GL(n), allowing weaker non-criticality conditions. I will start by giving a brief introduction to p-adic L-functions, and if time allows, will say a few words about our second main result, the variation of this construction in p-adic families.

18 November Matthew Bisatt (University of Bristol)
25 November Kyle Pratt (University of Oxford)
2 December Pip Goodman (University of Bristol)