# Thomas Lanard

Ph.D. student at IMJ-PRG
Supervisor: Jean-François DAT
Team: Formes Automorphes

Research Areas: Representation theory, Number theory, Algebra
Current interests: Representations of p-adic groups, Langlands program

## Publications

• Sur les $$\ell$$-blocs de niveau zéro des groupes $$p$$-adiques, Compos. Math. 154 (2018), no. 7 (pdf)
Let $$G$$ be a $$p$$-adic group that splits over an unramified extension. We decompose $$Rep_{\Lambda}^{0}(G)$$, the abelian category of smooth level $$0$$ representations of $$G$$ with coefficients in $$\Lambda=\overline{\mathbb{Q}}_{\ell}$$ or $$\overline{\mathbb{Z}}_{\ell}$$, into a product of subcategories indexed by inertial Langlands parameters. We construct these categories via systems of idempotents on the Bruhat-Tits building and Deligne-Lusztig theory. Then, we prove compatibilities with parabolic induction and restriction functors and the local Langlands correspondence.
• Sur les $$\ell$$-blocs de niveau zéro des groupes $$p$$-adiques II (pdf,arXiv:1806.09543)
Let $$G$$ be a $$p$$-adic group which splits over an unramified extension and $$Rep_{\Lambda}^{0}(G)$$ the abelian category of smooth level 0 representations of $$G$$ with coefficients in $$\Lambda=\overline{\mathbb{Q}}_{\ell}$$ or $$\overline{\mathbb{Z}}_{\ell}$$. We study the finest decomposition of $$Rep_{\Lambda}^{0}(G)$$ into a product of subcategories that can be obtained by the method introduced by Lanard, which is currently the only one available when $$\Lambda=\overline{\mathbb{Z}}_{\ell}$$ and $$G$$ is not an inner form of $$GL_n$$. We give two descriptions of it, a first one on the group side à la Deligne-Lusztig, and a second one on the dual side à la Langlands. We prove several fundamental properties, like for example the compatibility to parabolic induction and restriction or the compatibility to the local Langlands correspondence. The factors of this decomposition are not blocks, but we show how to group them to obtain "stable" blocks. Some of these results support a conjecture given by Dat.

• Thesis : Sur les $$\ell$$-blocs de niveau zéro des groupes $$p$$-adiques (pdf)

## Education

2015–

PhD in Mathematics,
Université Pierre et Marie Curie, IMJ-PRG, Paris

Subject: Blocks of p-adic groups and Langlands functoriality

2014-2015

Master (M2) of Mathematics,
École Normale Supérieure de Lyon

Subject: Introduction to the theory of L and zeta functions and their applications

2013-2014
Agrégation of Mathematics,
École Normale Supérieure de Lyon
Rank : 1st
2012-2013
Master (M1) of Mathematics, ERASMUS,
Imperial College London / École Normale Supérieure de Lyon
2011-2012
Bachelor of Mathematics,
École Normale Supérieure de Lyon

## Talks

February 2018
Sur les l-blocs des groupes p-adiques,
Séminaire Groupes Réductifs et Formes Automorphes de l’IMJ-PRG, Paris
November 2017
Sur les l-blocs des groupes p-adiques,
Séminaire Groupes, Représentations et Géométrie, Paris

## Research Internships

2015
Institut Mathématique de Jussieu, Jean François Dat
Representations of p-adic groups and Langlands conjectures
2013
Imperial College London, Kevin Buzzard
The Modularity Theorem
2012
École Polytechnique, Centre de Mathématiques Laurent Schwartz, Alain Plagne
Waring’s Problem

## Teaching

2015-2019
Mathematics in continuing education
L3, Autonomous
Polytech’ Paris
2018-2019
Group theory
L3, Teaching assistant of B. Stroh
Sorbonne Université
2018-2019
Arithmetic
L2, Teaching assistant of S. Chemla
Sorbonne Université
2015-2018
Fourier analysis and Distributions
L3, Teaching assistant of L. Lazzarini
Polytech’ Paris