Joint meeting, December 3-4 2025, Zürich
Speakers:
Timothée Bénard (CNRS, Sorbonne Paris Nord)
Tariq Osman (ETH Zürich)
Menny Aka (ETH Zürich)
Philippe Michel (EPFL)
Alexandru Pascadi (Bonn University)
December 3
- 12:00-13:30: Lunch at
- 13:30-15:00: talk by Dan Petersen in the Algebraic Geometry and Moduli Seminar Moments in families of L-functions over function fields via homological stability of moduli spaces
- 15:30-16:30: Menny Aka Distribution of planes in the Minkowski space
- 17:00-18:00: Tariq Osman Limit Theorems for Smooth Weyl Sums Associated to Quadratic Forms
- 19h: Dinner at
Rational subspaces of integral quadratic spaces, together with their invariants, encode rich arithmetic information: many classical and more recent arithmetic objects can be parametrized in this way. This perspective produces natural couplings between arithmetic objects that may at first look unrelated. It then leads to coupled distribution questions, to the search for arithmetic interpretations of such couplings, and to further applications of these relationships.
I will first illustrate this point of view by surveying earlier results, motivating the study of two-dimensional rational subspaces in Minkowski four-space. I will then describe the arithmetic invariants of such subspaces and the associated data.
Concretely, to each rational plane we associate a quadruple consisting of two closed geodesics in a Bianchi orbifold, one CM point on the modular surface, and one closed geodesic on the modular surface. We conjecture that, when these quadruples are grouped according to an associated discriminant, they jointly equidistribute, and we prove this under a congruence condition at one arbitrary prime. The proof adapts techniques from earlier work, with two main new ingredients: a general construction of the so-called Klein vectors and the use of variants of Duke’s theorem over imaginary quadratic fields. This is joint work with Konstantin Andritsch and Andreas Wieser.
Given a quadratic form Q in d variables, and fixed Schwartz function f, we may consider the associated smooth Weyl sum S^Q_N(t) := \sum_{n \in Z^d} f(N^{-1} n) e^{\pi i Q(n) t}, where t is a real number in the interval [0,1]. Such sums arise naturally when studying a certain non-local statistic of Q(Z^d), known as the spectral form factor. We discuss how dynamical methods can be used to prove the existence of the limit distribution for appropriately normalised Weyl sums when Q is a generic quadratic form, and also when Q is a generic diagonal quadratic form. This is part of work in progress with J. Griffin and J. Marklof.
- 9:30-10:30: Alexandru Pascadi Non-abelian amplification and bilinear forms with Kloosterman sums
- 11:00-12:00: Timothée Bénard Khintchine's theorem for fractal measures
- 12:00-13:30: Lunch at
- 13:30-14:30: Philippe Michel Bilinear sums with trace functions
We introduce a new approach to bound bilinear (Type II) sums of Kloosterman sums with composite moduli; combining this with previous results for prime moduli, we achieve savings beyond the Pólya-Vinogradov range for all moduli. We use Fourier analysis on SL2(Z/cZ), an amplification argument with non-abelian characters, and arrive at a counting problem related to the equidistribution of word values in PSL2(Z/cZ). We mention applications to moments of twisted cuspidal L-functions and to large sieve inequalities for exceptional cusp forms.
Khintchine's theorem is a key result in Diophantine approximation. Given a positive non-increasing function f defined over the integers, it states that the set of real numbers that are f-approximable has zero or full Lebesgue measure depending on whether the series of terms f(n))_n converges or diverges. I will present a recent work in collaboration with Weikun He and Han Zhang in which we extend Khintchine's theorem to any self-similar probability measure on the real line. The result provides an answer to an old question of Mahler, also asked by Kleinbock-Lindenstrauss-Weiss. The argument involves the quantitative equidistribution of upper triangular random walks on SL_2(R)/SL_2(Z).
We obtain non-trivial bounds for bilinear sums involving the trace function of an ell-adic sheaf, which go well below the Polya-Vinogradov range, under simple structural assumptions on its geometric monodromy group (for instance being simple); this is in contrast to previous works which required rather precise informations on the sheaf (such as the shape of its local monodromies at specific points along with global informations). Our approach is « soft » and builds on the method of moments of J. Xu, a robust version of the Goursat-Kolchin-Ribet criterion and on Sawin’s quantitative sheaf theory. In this talk we will describe this approach along with some applications. This is joint work with Etiene Fouvry, Emmanuel Kowalski and Will Sawin.