## Workgroup "Automorphic forms"

All the talks will take place at 9h30 in Kampé de Feriet room, unless otherwise stated.

## Spring 2022: Theories of local newforms

The purpose of this term's workgroup is to understand what has been done in the theory of local newforms since Casselman's seminal work on GL(2) and explore the various settings in which such results flourished, to hopefully understand deeper what could or should happen in general. All the references can be found here.

**March 24**- D. Lesesvre,*The theories of local newforms*, Notes

I briefly introduce the aims and spirit of the theories of newforms and the typical results we will be exploring during the workgroup.

**March 24**- J. Hauseux,*Casselman and GL(2)*, Notes

We discuss the paper of Casselman, proving the existence of newforms for GL(2), the associated multiplicity one theorem and discussing the relation with the classical L-functions and the notion of conductor of a representation. We introdude the necessary background before move towards the main arguments of the proofs.

**March 31**- M. Dimitrov,*Bernstein-Zelevinsky classification*, Notes

This talk will recall and present some essential results on supercuspidal representations, and will culminate with the classification of irreducible representations of GL(n).

**April 7**- D. Lesesvre,*JPSS and GL(n), part 1*, Notes

I will introduce many tools and ideas at the heart of the paper of Jacquet-Piatetski-Shapiro and Shalika, establishing the theory of local newforms on GL(n). The good properties of Whittaker functions make them have a "Fourier transform" that is polynomial in certain parameters. The existence (and other properties) of newforms can then be rephrased as properties on polynomials, and this point of view is essentially enough to prove all the results.

**April 14**- C.-Y. Hsu,*JPSS and GL(n), part 2*, Notes

We present the main ingredients of the proof of JPSS main result and its consequences.

**April 28**- Abhinandan,*Miyauchi and U(3), Notes*

In this talk we will discuss Miyauchi's proof of existence and multiplicity one property of newforms for an irreducible admissible generic representation of the unramified unitary group in three variables U(2,1) defined over a p-adic field.

**May 11**- N. Matringe,*Atobe, derivatives of representations and the non-generic case*

We present the notion of derivative of representation, that has been used in our paper to correct the proof of JPSS for GL(n) generic representations, and in Atobe's recent paper to give an analogous result in the case of non-generic representations of GL(n).

**June 9**- M. Dimitrov, Conductors of certain non-tempered representations of U(3)

I discuss some interesting consequences of Miyauchi's paper.

**June 30**- N. Raulf, Relations with the classical theory of newforms, of Atkin and Lehner?

Abstract to come

**Later?**- D. Lesesvre, Tsai and SO(2n+1)

Following a conjecture of Serre and Brumer, Tsai in her thesis addressed the first steps towards a theory of local newforms for SO(2n+1). She proves the existence of newforms in I will try to find a path inside this wide work to underline the most critical ingredients and emphasize the strong analogy with the seminal strategy of Jacquet, Piatetski-Shapiro and Shalika. Some fresh news will be on the table, by Cheng.

- Corrections to JPSS by Jacquet or Matringe
- Lansky-Raghuram SL(2) and U(2)
- Miyauchi U(3), and examples of non-generic cases
- Roberts-Schmidt GSp(4)
- Atobe for non-generic GL(n)
- Tsai and Lachaussée for SO(2n+1)

## Winter 21-22: Automorphic forms and L-functions

The purpose of this workgroup is to study some aspects of automorphic forms and their L-functions. This winter 2021-2022, we are aiming at introducing the theoretical setting for automorphic forms (representations, spectral theory, local and global aspects, ...) and advancing towards integral representations of L-functions.

**September 30**- M. Dimitrov,*Automorphic representations*, Notes

In this introductory talk of the reading seminar, assuming some familiarity with modular forms, we will present the representation theoretic framework where automorphic forms live

**October 7**- S. Jana,*Integral representations of L-functions*, Notes

We will start by talking about the Riemann zeta function and how it can almost be written as a Mellin integral of a nice function called theta series. Then we will try to give a big picture on what one can do for higher rank L-functions. In the process we will have ideas on automorphic forms and their Fourier expansions.

**October 14**- F. Brumley,*Classification of representations and automorphic forms*, Notes

We shall discuss the representation theory of SL2(R), linking it with the classical theory of automorphic forms on the upper half-plane.

**October 21**- R. Toma,*Spectral theory and decomposition of L^2*, Notes

We sketch the decomposition of the right-regular representation L^2(SL(2, Z) \ SL(2, R)). Doing so we introduce the theory of Eisenstein series with their meromorphic continuation and functional equation. Time permitting, we relate this decomposition to the spectral theory of the hyperbolic Laplacian, and make some remarks about how the decomposition generalises to congruence subgroups and then to higher rank groups like SL(3, R).

**November 12 at 2pm**- A. Betina,*Katz’ construction of the Eisenstein measure from an automorphic viewpoint*, Notes

We will talk about Eisenstein series and measures. They are very important to study integrality of Hecke L-values, but often come with a heavy load of computations. Harris, Hsieh, Li et al. began studying these Eisenstein series using automorphic forms instead, reducing the computations to simpler studies of zeta integrals, Mellin transform, etc. We will present how to estimate Eisenstein series for GL(2) at CM points with this approach.

**December 2**- E. Assing,*Whittaker models - An analytic perspective*, Notes

We will define the notion of a Whittaker model in the local setting and discuss the global Whittaker period of an automorphic form. To provide some practical context to these (technical) notions we will sketch how to derive the Whittaker expansion for some (generic) automorphic forms. Finally we will indicate how this expansion can be useful for certain estimates.

**January 20 and 27**- N. Raulf,*Archimedean aspects and (g,K)-modules*, Notes

In this talk we will consider the relation between representation theory and the classical theory of automorphic forms. We will review the basic ideas of Lie theory needed to treat the spectral decomposition in the cocompact case. If time permits we will also introduce (g, K)-modules to avoid analytic problems.

**February 3 and 10**- D. Lesesvre,*Whittaker models of automorphic representations*, Notes

In this talk we will start from the Fourier expansion of a classical automorphic cusp form on GL(2), and deduce by induction the Fourier expansion on GL(n). The Fourier coefficients involved in this expansion have particularly interesting properties, and constitute a privileged*model*of the representation. This model has an analogous meaning in the local setting, where they can be proven to be unique. From this very deep result, we deduce many nontrivial consequences for global models and automorphic forms.

**March 10**- D. N. Nguyen,*Integral representations of L-functions (II)*, Notes

We introduce the Rankin-Selberg integral representations of L-functions of automorphic forms, generalizing the integral representation of the Riemann zeta function. This representations allows to uncover the key properties of L-functions: they extend meromorphically on the whole complex plane, are bounded in vertical strips and satisfy a function equation relating s and 1-s.