Automorphic Forms Summer School
The theory of automorphic forms is a central area of modern mathematics due to its depth and connections to a large number of other disciplines. This feature makes it attractive but also difficult to enter. The summer school will provide a motivated introduction to this rich field through exciting recent developments. It is aimed at graduate students and early career researchers. Four minicourses will be given by top researchers of the younger generation. Financial support will be provided to selected participants in the form of accommodation and catering (details to be shared later).
- Jasmin Matz (University of Copenhagen): Equidistribution in automorphic families
Abstract: Automorphic forms and representations often naturally appear in families, for example, the family of all Hecke characters of a fixed number field, the family of holomorphic modular forms of growing level, or the family of Hecke-Maass forms of growing Laplace eigenvalue. It thus often makes sense to study families instead of single forms or representations, as this allows for more analytic methods to be used, and thus often makes certain conjectures that are otherwise out of reach more accessible (at least their probabilistic versions).
In addition, such families are conjectured to have additional underlying structure, so-called symmetry types, which are supposed to govern the distribution of the spectral parameters in those families. In this minicourse I want to discuss several instances of families, equidistribution results (known and conjectured), and their connection to conjectures such as the (vertical) Sato-Tate Conjecture and the Katz-Sarnak Conjectures on low-lying zeros in families of automorphic L-functions.
Paul Nelson (IAS Princeton): Estimates for L-functions
Abstract: A recurring problem in analytic number theory is to give estimates for special values of L-functions, either individually ("subconvexity") or on average over a family ("moments"). This problem can be approached via the theory of automorphic forms and their integral representations. I will give an introduction to recent results pursuing this approach, such as those in
The main local issue is to choose, in a given automorphic representation, vectors that make the integral representation well-suited for estimation. The main global issue is to provide nontrivial estimates for integrals of automorphic forms, which has been addressed in certain cases via Ratner theory or the pretrace formula. In addressing both the local and global issues, a key role is played by microlocal analysis on Lie group representations and the invariant theory of Gan--Gross--Prasad pairs.
- Danylo Radchenko (ETH Zürich): Modular forms, sphere packing, and energy minimization
Abstract: In this minicourse I will explain some old and new geometric applications of modular forms, from the classical theory of theta functions of lattices to the recent works of Viazovska on the sphere packing problem and of Cohn-Kumar-Miller-Radchenko-Viazovska on universal optimality for the energy minimization problem in dimensions 8 and 24.
- Benjamin Schraen (Université Paris-Sud, Orsay): Introduction to the p-adic Langlands Program
Absctract : The goal of the lectures will be to give an introduction to the p-adic Langlands Program, through the case of the group GL2(Qp). The p-adic Langlands corresondence relates p-adic 2-dimensional representations of the absolute Galois group of Qp and some representations of the group GL2(Qp) on p-adic Banach spaces. I will explain how such a correspondence is constructed and what it is useful in the theory of (p-adic) modular forms.
Application is now open. Application deadline: May 31, 2022.
The Erdős Center covers full lodging for selected participants. We try to support as many participants as possible. Please indicate in the application form if you wish to apply for this support.
MSc and PhD students applying to the summer school should also ask their supervisor to send a short recommendation (typically a few sentences) to the contact e-mail address email@example.com.
Link to the registration web-form of the summer school: