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

## 22nd meeting, London.

The 22nd meeting of the LPNTS will take place in London at UCL, on 5th and 6th June. The theme is "The Arthur trace formula, automorphic forms, arithmetic manifolds and their homology". The meeting will start at 1100 on the 5th (with coffee 1000-1100 beforehand) and finish on the afternoon of the 6th.

The talks on Monday 5th will be in room 505, 25 Gordon Street in the Department of Mathematics, UCL. The talks on Tuesday 6th will take place at Roberts, room 309. The location is about 10 minutes walk from the Eurostar terminal. You can look at the map of UCL and nearby area at this link.

The schedule:

Monday June 5th.

1000-1100 Coffee
1100-1200 Olivier Taïbi (Imperial): "An introduction to the stabilisation of the trace formula and Arthur-Langlands packets."
1200-1400 Lunch
1400-1500 Colette Moeglin (Jussieu): "On some local aspects of Arthur's theory"
1500-1545 Coffee
1545-1645 Nicolas Bergeron (Jussieu): "On the cohomology ring of the universal K3 surface"
1700-1800 Drinks

Tuesday June 6th.

1000-1100 Philippe Michel (Lausanne): "Sums of Kloosterman sums and sums of L-functions"
1100-1130 Coffee
1130-1230 Haluk Şengün (Sheffield): "Cohomology of arithmetic groups and Fermat's Last Theorem"
1230-1430 Lunch
1500-1700 Cultural activity

Some abstracts:

O. Taïbi: "An introduction to the stabilisation of the trace formula and Arthur-Langlands packets."
Abstract: Automorphic representations of reductive groups over number field generalise the classical notion of modular forms or Maass forms which are eigenvectors for the Hecke algebra. The Arthur-Selberg trace formula and its variants have proved extremely useful to study general automorphic representations, and their relation to Galois representations ("the Langlands programme"). In this introductory talk aimed at PhD students and non- specialists, I will explain what it means to stabilise trace formulae, and why it is crucial to understand this relation precisely: - locally (over local fields of characteristic zero), with the local Langlands correspondence and the introduction of Arthur-Langlands packets of representations, and - globally (over number fields), with Arthur's multiplicity formula for automorphic representations.

Colette Moeglin: "On some local aspects of Arthur's theory"
Abstract: I will first recall some basic facts about the spectral side of the trace formula to motivate the formulation of the local A-packet that I will give. After having given this formulation, I will explain what is known specially in the real case and what we expect to be true.

N. Bergeron: "On the cohomology ring of the universal K3 surface."
Abstract: The Deligne decomposition theorem makes it possible to reduce the study of the cohomology groups of the universal K3 surface, or more generally of universal families of polarized hyperkähler varieties, to the study of certain spaces of automorphic forms. This makes it possible to prove a cohomological version of the generalized Franchetta conjecture due to O'Grady but also to better understand the ring structure on the cohomology of these universal families. This is a joint work with Zhiyuan Li.

P. Michel: "Sums of Kloosterman sums and sums of L-functions."
Abstract: This talk is a review a recent series of works by V. Blomer, E. Fouvry, E. Kowalski, myself, D. Milicevic, W. Sawin as well as R. Zacharias. We will describe various estimates on sums of Kloosterman sums (or more generally trace functions) proven using methods from $\ell$-adic cohomology and some of their applications to the study of analytic properties of character twists of L-functions on average over the family of Dirichlet characters of some large prime modulus.

H. Şengün: "Cohomology of arithmetic groups and Fermat's Last Theorem."
Abstract: We use two fundamental reciprocity conjectures in the Langlands Programme that involve the cohomology of arithmetic groups and derive an algorithmically testable criterion for a number field K which, if satisfied, implies the truth of asymptotic Fermat's Last Theorem over K. Most imaginary quadratic fields satisfy the criterion. This is joint work with Samir Siksek (Warwick).

Previous meetings: