November 7th, 2025 (Kraków)
Intro talk
13:15-14:15 Jan Vonk (Leiden University)
Title: Around the class number one problem
Abstract: As part of a systematic computational study of equivalence classes of binary quadratic forms, Gauss stated several conjectures on their class numbers. In this talk, I will discuss some recent results related to these conjectures, with a special emphasis on the more recent geometric viewpoint on such questions using non-split Cartan modular curves.
Main talk
14:30-15:30 Jan Vonk (Leiden University)
Title: p-adic height pairings of geodesics
Abstract: I will discuss a certain p-adic height pairing of real quadratic geodesics on modular curves. The motivation for studying this pairing comes from its relation to real quadratic (RM) singular moduli. I will discuss how the interpretation of this height pairing as a triple product period sheds light on the conjectures that were made when RM singular moduli were defined. This is joint work with Henri Darmon.
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.
May 5th, 2025 (Poznań)
11:00 - 12:00 Kathrin Maurischat (RWTH Aachen University)
Title: Explicit families of Ramanujan bigraphs
Abstract: Similar to the Lubotzky-Phillips-Sarnak construction of regular Ramanujan graphs from PSL(2), Ramanujan bigraphs arise as quotients of Bruhat-Tits trees for PU(3). I give insight into joint work with S. Evra, B. Feigon, and O. Parzanchevski. In particular, I discuss the spectral property of being Ramanujan for higher regular graphs, give explicit examples for bigraphs, and give some of their properties. Furthermore, I show the number theoretic construction.
14:00 - 15:00 Matija Kazalicki (University of Zagreb)
Title: Asymptotics of D(q)-pairs via binary quadratic forms
Abstract: Let q be a nonzero integer. A set of distinct nonzero integers {a, b} is called a D(q)-pair if ab + q is a perfect square. Recently, Adžaga, Dražić, Dujella, and Pethő (for squarefree q), and Badesa (in general) determined the asymptotic behavior of D(q)-pairs using Dirichlet L-functions. In this talk, I will present an alternative approach to this problem based on binary quadratic forms, which yields slightly more general results. This is joint work with Goran Dražić.
March 3rd, 2025 (Poznań)
11:00 - 12:00 Abhishek Saha (Queen Mary University of London)
Title: Holomorphic QUE for Siegel cusp forms: the case of Saito-Kurokawa lifts
Abstract: The Quantum Unique Ergodicity (QUE) conjecture was proved in the classical case of Maass forms on the upper-half plane by Lindenstrauss and Soundararajan. The analogous mass equidistribution statement for holomorphic cusp forms in the weight aspect is a theorem due to Holowinsky and Soundararajan. In this talk, I will discuss some recent joint work with Jesse Jaasaari and Steve Lester on the higher rank analogue of the result of Holowinsky and Soundararajan for the case of holomorphic Siegel cusp forms. Our main result establishes mass equidistribution for Saito-Kurokawa lifts (which are special types of Siegel cusp forms of degree 2) assuming the Generalized Riemann Hypothesis (GRH) . We also show that this implies the equidistribution of zero divisors of Saito-Kurokawa lifts. Time permitting, I will say a few words about generalisation to higher dimensions.
February 3rd, 2025 (Kraków)
15:00-16:00 Monica Nevins (University of Ottawa)
Title: Why p-adic numbers are better than real in representation theory
Abstract: The p-adic numbers, discovered over a century ago, unveil aspects of number theory that the real numbers alone can’t. In this talk, we introduce p-adic fields and their fractal geometry, and then apply this to the (complex!) representation theory of the p-adic group SL(2). We describe a surprising conclusion: that close to the identity, all representations are a sum of finitely many rather simple building blocks arising from nilpotent orbits in the Lie algebra.
Jan 13th, 2025 (Kraków)
Introductory talks:
11:00-12:00 Dominik Kwietniak (UJ)
Title: An introduction to Ratner's theorems on unipotent flows I.
Abstract: Abstract: Unipotent flows are well-behaved dynamical systems. In particular, Marina Ratner has shown that the closure of every orbit for such a flow is of a nice algebraic (or geometric) form. This is known as the Ratner Orbit Closure Theorem. The other two closely related results are the Ratner Measure-Classification Theorem and the Ratner Equidistribution Theorem. These theorems resolve long-standing conjectures in ergodic theory and have applications in number theory, particularly in proving density results for integer points on homogeneous varieties. Their strength comes from complete classification of orbit closures and invariant measures for a whole class of important flows, something rare in the theory of dynamical systems.
The two introductory lectures are intended for a fairly general audience. Our aim is to provide an elementary introduction to the subject, explain these important theorems and some of their consequences. We concentrate on examples that illustrate the theorems and applications. The lectures are based on excellent lecture notes "Ratner’s Theorems on Unipotent Flows" written by Dave Witte Morris, see https://arxiv.org/pdf/math/0310402
12:0-13:30 Lunch break
13:30-14:30 Dominik Kwietniak (UJ)
Title An introduction to Ratner's theorems on unipotent flows II
14:30-15:00 Coffee break
15:00-16:00 Alex Gorodnik (University of Zurich)
Title: Equidistribution of unipotent periods and randomness of counting functions
Abstract: We study the limiting behavior of measures supported on unipotent orbits and discuss quantitative estimates for correlations of these measures. As an application we analyze statistical properties of counting functions representing the number of solutions of multiplicative Diophantine problems and show that they behave like sums of independent random variables. This is a joint work with Björklund and Fregolli.
Dec 2nd, 2024 (Kraków)
Introductory talks
9:00-10:00 Mikołaj Frączyk (UJ) "Expander graphs and applications"
10:00-10:30 Coffee break
10:30-12:00 Jolanta Marzec-Ballesteros (AMU) "Introduction to automorphic representations and the associated L-functions"
12:00-13:00 Lunch break
Research talks
13:30-14:30 Gergely Harcos (Renyi Institute, Budapest)
Title: A new zero-free region for Rankin–Selberg L-functions
Abstract: I will present a new zero-free region for all GL(1)-twists of GL(m) × GL(n) Rankin–Selberg L-functions. The proof is inspired by Siegel’s celebrated lower bound for Dirichlet L-functions at s = 1. I will also discuss two applications briefly. Joint work with Jesse Thorner.
14:30-15:00 Coffee break
15:00 -16:00 Harald Helfgott (IMJ-PRG, Paris)
Title: Expansion, divisibility and parity
Abstract: We will discuss a graph that encodes the divisibility properties of integers by primes. We will prove that this graph has a strong local expander property almost everywhere. We then obtain several consequences in number theory, beyond the traditional parity barrier, by combining the main result with Matomaki-Radziwill. (This is joint work with M. Radziwill.) For instance: for lambda the Liouville function (that is, the completely multiplicative function with lambda(p) = -1 for every prime), (1/\log x) \sum_{n\leq x} lambda(n) \lambda(n+1)/n = O(1/sqrt(log log x)), which is stronger than well-known results by Tao and Tao-Teravainen. We also manage to prove, for example, that lambda(n+1) averages to 0 at almost all scales when n restricted to have a specific number of prime divisors Omega(n)=k, for any "popular" value of k (that is, k = log log N + O(sqrt(log log N)) for n<=N).
We shall also discuss a recent generalization by C. Pilatte, who has succeeded in proving that a graph with edges that are rough integers, rather than primes, also has a strong local expander property almost everywhere, following the same strategy. As a result, he has obtained a bound with O(1/(log x)^c) instead of O(1/sqrt(log log x)) in the above, as well as other improvements in consequences across the board.
Nov 4th, 2024 (Poznań) Seminar room B1-37
14:00 - 15:00 Rafail Psyroukis (Durham University)
Title: Analytic properties of Fourier-Jacobi Dirichlet series
Abstract: A Dirichlet series involving the Fourier-Jacobi coefficients of two Siegel modular forms has been extensively studied, mainly due to its connection with the spinor L-function. In this talk, we will consider a similar Dirichlet series, but for modular forms of orthogonal groups of signature (2, n+2) and focus on its analytic properties, i.e. its analytic continuation to the complex plane and functional equation. We will begin by providing an overview of the problem for Siegel modular forms, followed by key definitions of orthogonal modular forms, and conclude by outlining the main steps in our proof for this case.
Oct 7th, 2024 (Kraków) Conference room under the library
10:30-12:00 Anna Szumowicz (UJ) introductory lecture "Introduction to representations of p-adic groups"
12:00-13:30 lunch break
13:30-14:30 Arnaud Mayeux (Hebrew University of Jerusalem)
Title: Multi-centered dilatations and applications to congruent isomorphisms for group schemes
Abstract: Let X be a scheme and let Y and D be closed subschemes of X such that D is locally principal. The dilatation of X with center [Y,D] is an X-affine scheme enjoying many properties. Dilatations tend to preserve group schemes. This construction was studied and used by many authors, including Bosch-Lütkebohmert-Raynaud, Prasad-Yu, Dubouloz and M-Richarz-Romagny. In the talk, I will introduce dilatations, including a multi-centered version of this construction. Then, I will state and prove some congruent isomorphisms for group schemes. Congruent isomorphisms are related to Moy-Prasad isomorphisms, frequently used in representation theory of p-adic groups.
14:30-15:00 coffee break
15:00-16:00 Maarten Solleveld (Radboud Universiteit Nijmegen)
Title: Towards a local Langlands correspondence for depth zero representations
Abstract: Among the representations of a reductive p-adic group G, those in depth zero are the ones most closely related to representations of reductive groups over finite fields. We will discuss a local Langlands correspondence for irreducible depth zero G-representations (on complex vector spaces), that is considerably easier than in positive depth. We will see that for large classes of such representations, an explicit, constructible LLC is known. We will also look at some ideas on how to generalize this to all irreducible depth zero representations, which is joint work in progress with Tasho Kaletha.