Number Theory Seminar
Unless otherwise specified, the seminars are held on Mondays at 15:00 in Room B3.03 – Mathematics Institute
201920 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 padic etale cohomology of X_{K_p K^p} vanishes above the middle degree. 
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 nonempty subset of the coefficients sums to zero, which is a higherdegree 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) padic Lfunctions 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 DimitrovJanuszewskiRaghuram gives a padic Lfunction attached to pi, that is, a padic measure interpolating its classical critical Lvalues. I will report on ongoing joint work with Daniel Barrera and Mladen Dimitrov where we generalise this to the nonordinary 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 noncriticality conditions. I will start by giving a brief introduction to padic Lfunctions, and if time allows, will say a few words about our second main result, the variation of this construction in padic families. 
18 November  Matthew Bisatt (University of Bristol) 
25 November  Kyle Pratt (University of Oxford) 
2 December  Pip Goodman (University of Bristol) 