# Thomas Lanard

Postdoc at Vienna University, Austria

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

## Publications

• Modulo $$\ell$$ distinction problems for $$\mathrm{GL}_2$$ and $$\mathrm{SL}_2$$ (pdf)
with Peiyi Cui, Hengfei Lu
This paper concerns the $$\ell$$-modular representations of $$\mathrm{GL}_2(E)$$ and $$\mathrm{SL}_2(E)$$ distinguished by a Galois involution, with $$\ell$$ an odd prime different from $$p$$. We start by proving a general theorem allowing to lift supercuspidal $$\overline{\mathbf{F}}_{\ell}$$-representations of $$\mathrm{GL}_n(F)$$ distinguished by an arbitrary closed subgroup $$H$$ to a distinguished supercuspidal $$\overline{\mathbf{Q}}_{\ell}$$-representation. Then we give a complete classification of the $$\mathrm{GL}_2(F)$$-distinguished representations of $$\mathrm{GL}_2(E)$$, where $$E$$ is a quadratic extension of $$F$$. For supercuspidal representations, this extends the results of Sécherre to the case $$p=2$$. Using this classification we discuss a modular version of the Prasad conjecture for $$\mathrm{PGL}_2$$. We show that the "classic" Prasad conjecture fails in the modular setting. We propose a solution using non-nilpotent Weil-Deligne representations. Finally, we apply the restriction method of Anandavardhanan and Prasad to classify the $$\mathrm{SL}_2(F)$$-distinguished modular representations of $$\mathrm{SL}_2(E)$$.
• Depth zero representations over $$\overline{\mathbb{Z}}[\frac{1}{p}]$$ (pdf)
with Jean-François Dat
We consider the category of depth $$0$$ representations of a $$p$$-adic quasi-split reductive group with coefficients in $$\overline{\mathbb{Z}}[\frac{1}{p}]$$. We prove that the blocks of this category are in natural bijection with the connected components of the space of tamely ramified Langlands parameters for $$G$$ over $$\overline{\mathbb{Z}}[\frac{1}{p}]$$. As a particular case, this depth $$0$$ category is thus indecomposable when the group is tamely ramified. Along the way we prove a similar result for finite reductive groups. We then outline a potential application to the Fargues-Scholze and Genestier-Lafforgue semisimple local Langlands correspondences. Namely, contingent on a certain "independence of $$\ell$$" property, our results imply that these correspondences take depth $$0$$ representations to tamely ramified parameters.
• Unipotent $$\ell$$-blocks for simply connected $$p$$-adic groups (pdf)
Let $$F$$ be a non-archimedean local field and $$G$$ the $$F$$-points of a connected simply-connected reductive group over $$F$$. In this paper, we study the unipotent $$\ell$$-blocks of $$G$$. To that end, we introduce the notion of $$(d,1)$$-series for finite reductive groups. These series form a partition of the irreducible representations and are defined using Harish-Chandra theory and $$d$$-Harish-Chandra theory. The $$\ell$$-blocks are then constructed using these $$(d,1)$$-series, with $$d$$ the order of $$q$$ modulo $$\ell$$, and consistent systems of idempotents on the Bruhat-Tits building of $$G$$. We also describe the stable $$\ell$$-block decomposition of the depth zero category of an unramified classical group.
• Equivalence of categories between coefficient systems and systems of idempotents, Represent. Theory 25 (2021), 422-439 (pdf)
The consistent systems of idempotents of Meyer and Solleveld allow to construct Serre subcategories of $$Rep_{R}(G)$$, the category of smooth representations of a $$p$$-adic group $$G$$ with coefficients in $$R$$. In particular, they were used to construct level 0 decompositions when $$R=\overline{\mathbb{Z}}_{\ell}$$, $$\ell \neq p$$, by Dat for $$GL_n$$ and the author for a more general group. Wang proved in the case of $$GL_n$$ that the subcategory associated with a system of idempotents is equivalent to a category of coefficient systems on the Bruhat-Tits building. This result was used by Dat to prove an equivalence between an arbitrary level zero block of $$GL_n$$ and a unipotent block of another group. In this paper, we generalize Wang's equivalence of category to a connected reductive group on a non-archimedean local field.
• Sur les $$\ell$$-blocs de niveau zéro des groupes $$p$$-adiques II, Ann. Sci. Éc. Norm. Supér. (4) 54 (2021), no. 3, 683–750 (pdf)
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.
• 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.

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

## Education

2019–
Post-Doc in Mathematics,
University of Vienna, Austria
2015-2019

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

Subject: On the l-locks of p-adic groups

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

July 2022
AMS-SMF-EMS Joint International Meeting Special Sessions, Grenoble, France
June 2022
Seminar Number Theory of the ENS Lyon, Lyon, France
June 2022
Paris-London Number Theory Seminar, Londres, Royaume-Uni
May 2022
Seminar of the Laboratoire de Mathématiques de Versailles, Versailles, France
March 2022
Seminar Representation Theory and Automorphic Forms, Vienna, Austria
October 2021
Seminar Groupes, Algèbre et Géométrie, Poitiers, France
April 2021
Seminar Géométrie complexe, Online talk, Nancy, France
January 2021
Seminar Théorie des groupes, Online talk, Amiens, France
November 2020
Seminar Géométrie Arithmétique et Motivique, Online talk, Paris, France
December 2019
London Number Theory Seminar, London, United-Kingdom
February 2019
Colloquium GDR TLAG, Poitiers, France
February 2019
Seminare of the University of East Anglia, Norwich, United-Kingdom
December 2018
Seminare of the University of Vienna, Vienna, Austria
November 2018
Seminare of the Laboratoire de Mathématiques de Versailles, Versailles, France
February 2018
Seminare Groupes Réductifs et Formes Automorphes de l’IMJ-PRG, Paris, France
November 2017
Seminare Groupes, Représentations et Géométrie, Paris, France

## 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

2021-2022
Linear Algebra
L1, TD
University of Vienna
2019-2029
Algebraic Number Theory
Master
University of Vienna
2020-2021
Linear Algebra and Geometry
L1
University of Vienna
2020-2021
Number Theory
L1
University of Vienna
2015-2019
Mathematics in continuing education
L3, Autonomous
Polytech’ Paris
2018-2019
Group theory
L3
Sorbonne Université
2018-2019
Arithmetic
L2
Sorbonne Université
2015-2018
Fourier analysis and Distributions
L3
Polytech’ Paris