AF/NT seminar

Organizers: Mikołaj Frączyk (UJ), Jolanta Marzec-Ballesteros (UAM) and Anna Szumowicz (UJ)

Automorphic Forms/ Number Theory seminar is a recurrent meeting wandering between Jagiellonian University in Krakow (JU)  and Adam Mickiewicz University in Poznań (AMU), typically on the first Monday of the month. We plan to have two invited speakers per session preceded by  introductory talks by junior faculty or graduate students. 

We have limited funds to support travel for the participants from Poland to the seminar venue. 

Talks in Krakow will take place in the Conference Hall B  under the library at the Faculty of Mathematics and Computer Science of JU . Talks in Poznań will take place in the seminar room B1-37 at the Faculty of Mathematics and Computer Science of AMU. 

If you would like to join one of the talks online, contact one of the organizers to obtain a link.

Upcoming Meetings

October 10th, 2025 (Kraków) 

Intro talk 

14:00-14:50 Mikolaj Fraczyk (UJ) 

Title: Simple trace formula and some applications

Abstract: I will start by recalling the Poisson summation formula. We will see how to prove it using spectral theory of Laplace operator on a flat torus, and how this proof generalizes to trace formulas on compact manifolds. The we will discuss the shape of the Selberg trace formula for compact hyperbolic surfaces. 

Main talk

15:00-16:00 Tobias Finis (Leipzig University)

Title: Prehomogeneous vector spaces and the trace formula: results and examples

Abstract: Arthur's trace formula is a fundamental tool in the study of automorphic representations of a reductive group $G$ over a number field. In a first approximation, it expresses the duality between the spectrum and the conjugacy classes of the group of rational points of $G$. However, the picture is complicated by the contribution of the continuous spectrum to the spectral side and by the terms corresponding to non-elliptic conjugacy classes on the geometric side. The formula results from a truncation process, which produces weighted orbital integrals with geometric weight factors (volumes of polytopes) for semisimple conjugacy classes, but is in general more complicated for non-semisimple, in particular unipotent classes.

Is is known that the contribution of a geometric unipotent class can be expressed as the absolutely convergent integral of an alternating sum of terms, the first of which is a sum over the rational points of the orbit. One can try to use the description of geometric unipotent orbits by Dynkin and Kostant to obtain a more explicit expression. Here, prehomogeneous vector spaces (representations of reductive groups with an open orbit) arise naturally. In a number of cases the contribution of an orbit to the trace formula can be explicitly evaluated in terms of the zeta functions of these spaces, which generalize the classical Dedekind and Epstein zeta functions. We will discuss the definition of these zeta functions and a recent general convergence result. We will then return to Arthur's trace formula, review some transparent cases and some others, where the final description is more complicated. This is joint work with Erez Lapid.

December 5th, 2025 (Poznań) 

Didier Lesesvre (Université de Lille)