## Workgroup "Automorphic forms"

The purpose of this workgroup is to study some aspects of automorphic forms and their L-functions. This year 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.

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

## Scheduled talks

**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.

**December 16**- N. Raulf,*Archimedean aspects and (g,K)-modules*

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.

**January 6**- D. Lesesvre,*Whittaker models of automorphic representations*

Abstract to come

**January 13**- D. N. Nguyen,*Integral representations of L-functions (II)*

Abstract to come

## References

Specific references will be provided for each talk. The following books will be used as a guideline for the workgroup:

- Cogdell, Kim and Murty,
*Lectures on Automorphic L-functions*, Fields Institute Monographs, 2004 - Fleig et al.,
*Eisenstein Series and Automorphic Representations, with applications to String Theory*, Cambridge UP, 2018 - Bump,
*Automorphic forms and representations*, Cambridge UP, 1998