# The London-Paris Number Theory Seminar

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: Pierre Charollois, Olivier Fouquet, Michael Harris, Marc Hindry, Benjamin Schraen, Jacques Tilouine.

## 20th meeting, London.

The 20th meeting of the LPNTS will take place in London (at UCL), on 6th and 7th June 2016. The themes are representations of p-adic groups and arithmetic geometry.

Schedule:

06/06/16

Coffee/tea/pastries 10:00-11:00
Talk 1 11:00-12:00 Roessler
Lunch 12:00-14:00
Talk 2 14:00-15:00 Taelman
Coffee 15:00-15:45
Talk 3 15:45-16:45 Cadoret
Drinks 17:00-18:00

06/07/16

Talk 4 9:30-10:30 Stevens
Coffee 10:30-11:00
Talk 5 11:00-12:00 Aubert
Lunch 12:00-14:00
Talk 6 14:00-15:00 Le Bras
Coffee 15:00-15:45
Talk 7 15:45-16:45 Ardakov

Here are the titles and abstracts which I know about:

Shaun Stevens (UEA)
Title: Representations of p-adic groups and the local Langlands correspondence
Abstract: In this introductory talk, I will try to describe the some of the ideas, techniques, and questions in the representation theory of p-adic groups, motivated by the local Langlands correspondence, mostly just for complex representations. I will assume some familiarity with local fields and with the representation theory of finite groups.

Damian Roessler (Oxford)
Ttitle: Strongly semistable sheaves and the Mordell-Lang conjecture over function fields.
Abstract: We shall describe a proof of a strengthening of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety.
If $K_0$ is a function field in one variable over a finite field $k_0$ of char. $p>0$ and $X$ a smooth closed subvariety of an abelian variety $A$ over $K_0$, then we show that if $X(K_0)$ is dense in $X$ and $p>{\rm dim}(X)^2{\rm deg}(\Omega_X)$, then $X_{K^{\rm sep}_0}$ must be defined over $\bar{k_0}$. Here the degree is defined with respect to a fixed but arbitrary ample line bundle on $X$.
This is a strengthening of the usual Mordell-Lang conjecture for $X$, because the latter only predicts that $X$ is isotrivial up to a finite purely inseparable morphism.
Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on Langer's theorem that the Harder-Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs.

Anna Cadoret (Ecole Polytechnique)
Title: Geometric monodromy - semisimplicity and maximality
Abstract: Let X be a connected scheme, smooth and separated over an algebraically closed field k of characteristic p, let f:Y-> X be a smooth proper morphism and x a geometric point on X. We show that the tensor invariants of bounded length d of the etale fundamental group \pi_1(X) acting on the \'etale cohomology groups H^*(Y_x,\F_l) are the reduction modulo-l of the tensor invariants of \pi_1(X,x) acting on H^*(Y_x,\Z_l) for l large enough depending on f:Y-> X, d. We use this result to discuss semisimplicity and maximality issues about the image of \pi_1(X,x) acting on H^*(Y_x,\Z_l). This is a joint work with Chun-Yin Hui and Akio Tamagawa.

Arthur-César Le Bras (Ecole Normale)
Title : Drinfeld’s coverings and the $p$-adic Langlands correspondence
Abstract : I will explain the proof of (a version of) a conjecture of Breuil-Strauch, which gives a purely geometric description of the $p$-adic local Langlands correspondence for $GL_2(\mathbf{Q}_p)$ for de Rham non trianguline Galois representations, using Drinfeld's coverings of the $p$-adic upper half-plane. It can be seen as a $p$-adic analogue of the realization of the (classical) local Langlands correspondence for supercuspidal representations in the $\ell$-adic cohomology of the Drinfeld tower. This is a joint work with Gabriel Dospinescu.

Lenny Taelman (Amsterdam)
Title: Complex multiplication and K3 surfaces over finite fields
Abstract: The zeta function of a K3 surface over a finite field satisfies a number of obvious (archimedean and l-adic) and a number of less obvious (p-adic) constraints. We consider the converse question, in the style of Honda-Tate: given a rational function Z satisfying all these constraints, does there exist a K3 surface whose zeta-function equals Z?

Konstantin Ardakov (Oxford)
Title: Admissible representations of p-adic Lie groups, and D-modules
Abstract: The localisation theorem of Beilinson and Bernstein opened new doors in representation theory by relating representations of semisimple Lie algebras to D-modules on flag varieties. I will talk about an extension of this result, which gives an anti-equivalence between the category of admissible representations (locally analytic, with trivial infinitesimal central character) of a semisimple p-adic Lie group and the category of coadmissible equivariant D-modules on the rigid analytic variety.

Anne-Marie Aubert (Jussieu)
Title: TBA
Abstract: TBA

Note: some of these talks are introductory talks, aimed at PhD students.

The meeting will start at 11:00 on June 6 and finish at 16:45 on June 7. The talks will be in room 505, 25 Gordon Street in the Department of Mathematics, UCL. The location is about 10 minutes' walk from the Eurostar terminal. You can look at the Map of UCL and nearby area here. Previous few meetings:

This page is maintained by Kevin Buzzard.