# Algebra Seminar

- This event is offered only in English.

SPEAKER: Peter Latham (King's College London)

A lot of interesting arithmetic information is contained in congruences between l-adic representations of the absolute Galois group of a p-adic field, for l and p rational primes. One approach to studying when this can happen is via deformation theory, which allows one to use methods of algebraic geometry, and particularly intersection theory, to study such congruences. A conjecture of Breuil and Mézard predicts that the intersection theory of the associated geometric “deformation spaces” can be described in terms of the representation theory of GL(n,Z_p), and this conjecture is intimately related to central problems in algebraic number theory – for example, Kisin’s proof of the 2-dimensional Fontaine—Mazur conjecture follows from establishing the Breuil—Mézard conjecture for 2-dimensional representations when l=p. More recently, Shotton has proved the Breuil—Mézard conjecture in complete generality in the case that l and p are distinct. I will describe work extending Shotton’s result to fit in with the wider context of the local Langlands programme, leading to a description of congruences between the “tame regular semisimple elliptic Langlands parameters” of DeBacker and Reeder in terms of the representation theory of certain p-adic integral group schemes.