### Seminars/GdT - Working groups or reading groups to which I participate.

#### Théorie de Hodge des morphismes projectifs, d'après de Cataldo et Migliorini (Fall 2023)

#### Variétés abéliennes et théorème de Honda-Tate (Fall 2016, organiser Giuseppe Ancona)

Here are liveTeX notes (french) that I took (last update 25 Jan 2017). I'm responsible for all the errors.

## Research interests

I did my PhD with Olivier Schiffmann at Universite Paris 7 and I defended it in July 2014. My research area is geometric representation theory.

I've worked on the Hall algebra of an elliptic curve and on the spherical Eisenstein sheaves for an elliptic curve.

- Geometric Langlands correspondence and applications
- Eisenstein sheaves, elliptic character sheaves
- Automorphic forms for functions fields
- Principal bundles and their moduli stacks/spaces
- Representations of algebraic groups and p-adic groups
- Algebraic cycles, periods and p-adic periods
- Hall algebras for quivers/curves

### Papers

Disclaimer and warning: the pdf linked to are not identical to the published version.- Ngô support theorem and polarizability of quasi-projective commutative group schemes
- Algebraic classes in mixed characteristic and André's p-adic periods
- Homological duality for covering groups of reductive p-adic groups
- The Jordan--Chevalley decomposition for G-bundles on elliptic curves
- Revisiting the moduli space of semistable G-bundles over an elliptic curve
- On the stack of semistable G-bundles on an elliptic curve

Published version Math. Annalen, 2016, 365:401–421. - Addendum to Olivier Schiffmann:"On the Drinfeld realization of the elliptic Hall algebra"

Published version Journal of Algebraic Combinatorics March 2012, Volume 35, Issue 2, pp 263-267 - Cusp eigenforms and the Hall algebra of an elliptic curve

Published version Compositio Math. Vol 149 Issue 06 June 2013, pp 914-958 - Crossed product of cyclic groups

joint with Ana-Loredana Agore

Published version Czecholslovak Mathematical Journal December 2010, Volume 60, Issue 4, pp 889-901

(joint with Giuseppe Ancona) We prove that any commutative group scheme over an arbitrary base scheme of finite type over a field with connected fibers and admitting a relatively ample line bundle is polarizable in the sense of Ngô. This extends the applicability of Ngô's support theorem to new cases, for example to Lagrangian fibrations with integral fibers and has consequences to the construction of algebraic classes.

(joint with Giuseppe Ancona) Motivated by the study of algebraic classes in mixed characteristic we define a countable subalgebra of ℚ_p which we call the algebra of André's p-adic periods. We construct a tannakian framework to study these periods. In particular, we bound their transcendence degree and formulate the analog of the Grothendieck period conjecture. We exhibit several examples where special values of classical p-adic functions appear as André's p-adic periods and we relate these new conjectures to some classical problems on algebraic classes.

(joint with Dipendra Prasad) In this largely expository paper we extend properties of the homological duality functor RHom_H(,-H)where H is the Hecke algebra of a reductive p-adic group, to the case where it is the Hecke algebra of a finite central extension of a reductive p-adic group. The most important properties being that RHom_H(−,H) is concentrated in a single degree for irreducible representations and that it gives rise to Schneider--Stuhler duality for Ext groups (a Serre functor like property). Along the way we also study Grothendieck--Serre duality with respect to the Bernstein center and provide a proof of the folklore result that on admissible modules this functor is nothing but the contragredient duality. We single out a necessary and sufficient condition for when these three dualities agree on finite length modules in a given block. In particular, we show this is the case for all cuspidal blocks as well as, due to a result of Roche, on all blocks with trivial stabilizer in the relative Weyl group.

(joint with Sam Gunningham and Penghui Li) We study the moduli stack of degree 0 semistable G-bundles on an irreducible curve E of arithmetic genus 1, where G is a connected reductive group. Our main result describes a partition of this stack indexed by a certain family of connected reductive subgroups H of G (the E-pseudo-Levi subgroups), where each stratum is computed in terms of H-bundles together with the action of the relative Weyl group. We show that this result is equivalent to a Jordan--Chevalley theorem for such bundles equipped with a framing at a fixed basepoint. In the case where E has a single cusp (respectively, node), this gives a new proof of the Jordan--Chevalley theorem for the Lie algebra 𝔤 (respectively, group G). We also provide a Tannakian description of these moduli stacks and use it to show that if E is an ordinary elliptic curve, the collection of framed unipotent bundles on E is equivariantly isomorphic to the unipotent cone in G. Finally, we classify the E-pseudo-Levi subgroups using the Borel--de Siebenthal algorithm and compute some explicit examples.

The main result says that we can describe the moduli space (not stack) of G-bundles of arbitrary degree on an elliptic curve in terms of line bundles and a certain Weyl group. The idea is to use the previous result that describes the Jordan-Hölder "series" of a G-bundle on an elliptic curve and some cohomological computations. The proof is algebraic and works in arbitrary characteristic. Important work has been previously done by Laszlo and Friedman-Morgan (and here). Our method is close in spirit to Laszlo's.

This paper grew from an attempt to understand the simple summands of spherical Eisenstein sheaves for an elliptic curve. The main result is that for every component of the stack of G-bundles on an elliptic curve there is a unique smallest parabolic subgroup such that the induction map is a small map and the deck group is proved to be a certain Weyl group. This extends a result of Ben-Zvi and Nadler.

This is a short note that is not self contained at all in which I prove that the presentation of the Drinfeld double of the elliptic Hall algebra from the paper in the title actually holds for the half, i.e. for the elliptic Hall algebra itself. The proof is just some combinatorial trick and abstract non-sense of the Drinfeld double construction. The result can be interpreted as a description of higher (more than quadratic) functional equations of the Eisenstein series for an elliptic function field.

I give an explicit construction of the cusp eigenforms on an elliptic curve defined over a finite field using the theory of Hall algebras and the Langlands correspondence for function fields and GL(n). As a consequence we obtain a description of the Hall algebra of an elliptic curve as an infinite tensor product of simpler algebras. We prove that all these algebras are specializations of a universal elliptic Hall algebra defined by Burban and Schiffmann.

We describe explicitly by generators and relations all the crossed products of two cyclic groups. A necessary and sufficient condition for an extension of a group by a group to be a cyclic group is given. This can be viewed as an extension of the Chinese Remainder Theorem for non coprime integers.