Thomas Lanard

About

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)

    CV

    Pdf version

    Education

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

    PhD in Mathematics,
    Advisor : Jean-François Dat,
    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