## 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.

## CV

## 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*