The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by grants from ANR Projet ArShiFo ANR-BLAN-0114, EPSRC Platform Grant EP/I019111/1, PerCoLaTor (Grant ANR-14-CE25), the Heilbronn Institute for Mathematical Research, and ERC Advanced Grant AAMOT.

London organizers: David Burns, Kevin Buzzard Fred Diamond, Yiannis Petridis, Alexei Skorobogatov, Andrei Yafaev, Sarah Zerbes.

Paris organizers: Matthew Morrow, Olivier Fouquet, Michael Harris, Marc Hindry, Benjamin Schraen, Jacques Tilouine.

The 24nd meeting of the LPNTS took place in London at UCL, on 29th and 30th of May, 2018. The theme was "mod p automorphic forms".

The schedule was

Tuesday 29th:

1000--1100: Coffee

1100--1200: Payman Kassaei

1200--1400: lunch

1400-1500: Marie-France Vigneras

1500-1530: coffee

1530-1600: David Helm

1630-1730: Stefano Morra

Wednesday 30th:

0900-1000: Julien Hauseux

1000-1030: coffee

1030-1130: Pascal Boyer

1130-1230: Jack Thorne

Titles and abstracts:

Payman Kassaei (King's College London)

Title: Stratifications on mod p Shimura varieties and applications to automorphic forms

Abstract: In this talk, I will describe in an introductory way stratifications on certain mod p Shimura varieties and present some past and recent applications to arithmetic of automorphic forms.

Marie-France Vigneras (Jussieu)

Title: On supersingularity

Abstract: There are explicit relations between:

Irreducible admissible representations of $GL_2(Q_p)$ with supersingular reduction modulo $p$,

Supersingular simple modules modulo $p$ of the pro-$p$ Iwahori Hecke algebra of $GL_2(Q_p)$,

Irreducible 2-dimensional representations modulo $p$ of the Galois group of $Q_p$.

How does this generalize to a finite extension $F/Q_p$, a positive integer $n$, a reductive $p$-adic group?

Pascal Boyer (Paris 13)

Title: Torsion or not Torsion

Abstract: About the $\mathbb Z_l$-cohomology of Shimura varieties, on can be interested by killing the torsion
using for example localization at some well chosen maximal ideal of some Hecke algebra. On the opposite point of
view, we can ask for the arithmetic meaning of torsion classes so that we are led to the problem of the construction
of such classes. In this talk we will try to tackle these two aspects in the particular case of Shimura varieties of
Kottwitz-Harris- Taylor type.

Julien Hauseux (Lille)

Title: Extensions between generalised Steinberg representations

Let G be a p-adic reductive group. We compute the extensions between mod p smooth generalised Steinberg
representations of G. This is part of a work in progress with Colmez, Dospinescu, and Nizioł.

David Helm (Imperial College)

Title: Towards a local Langlands correspondence in families for split groups in depth zero

Abstract: The local Langlands correspondence in families identifies the integral Bernstein centre for the group
GL_n(F) with a certain ring of functions on the moduli space of Langlands parameters for GL_n. I will describe a
conjectural generalization of this result to depth zero representations of a split reductive group G over F, and
explain how a key part of the proof of local Langlands in families for GL_n generalizes to this case. This is joint
work with Jean-Francois Dat, Rob Kurinczuk, and Gil Moss.

Stefano Morra (Montpellier)

Title: Local models for Galois representations, and applications to automorphic forms

Deformation spaces of finite at group schemes are a central subject in p-adic Hodge
theory, and in arithmetic questions of a global nature. The description of their singularities
in specific situations led to a wide horizon of achievements, from the proof of the
conjectures of Serre (as generalized by Buzzard-Diamond- Jarvis, Schein, Gee and others)
to the establishment of the Shimura-Taniyama- Weil conjecture and many more cases of
modularity lifting theorems in dimension 2.
Nevertheless outside a limited number of situations (Barsotti-Tate, ordinary, Fontaine-
Laffaille) the geometry of more general Galois deformation spaces remains mysterious. In
this talk we introduce an algebraic variety, refinement with monodromy of the local model
of Pappas-Rapoport, which controls the structure of generic potentially crystalline
deformation spaces of Galois representations in arbitrary dimension. In particular we
illustrate how its geometry is predicted by and predicts generalized geometric
interpretations of the weight part of Serre and Breuil-Mézard conjectures as proposed by
Emerton and Gee. This is ongoing joint work with Daniel Le, Bao Viet Le Hung and
Brandon Levin.

Jack Thorne (Cambridge)

Title : Odd Galois Representations

Abstract : A popular slogan is that "all Galois representations which
appear in the cohomology of Shimura varieties are conjugate self-dual up
to twist". However, fine words butter no parsnips, and this is on the
face of it not quite true. I will discuss what one can say in this
direction. This is joint work with Christian Johansson.

Previous meetings:

18th meeting (Imperial, 4--5/6/15)

19th meeting (Paris 13, 9/11/15)

20th meeting (UCL, 6--7/6/16)

21st meeting (Jussieu, 14--15/11/16)

22nd meeting (UCL, 5--6/6/17)

23rd meeting (Jussieu, 27--28/11/17)

This page is maintained by Kevin Buzzard.