Number Theory Day
Luxembourg Number Theory Day 2025
This year, the Luxembourg Number Theory Day will be held on Friday, December 19. The talks will be held in room MSA 3.330 of the Maison du Savoir on the Esch-Belval campus of the University of Luxembourg.
| Speaker | Title of the talk | Time |
|---|---|---|
| Lennart Gehrmann | An introduction to rigid meromorphic cocycles | 9:30-10:30 |
| Madhavan Venkatesh | Counting points on varieties over finite fields | 11:00-12:00 |
| Alexandre Maksoud | Families of p-adic modular forms and their geometry | 13:30-14:30 |
| Marusia Rebolledo | Beading workshop | 15:00-16:00 |
Abstracts
Lennart Gehrmann, An introduction to rigid meromorphic cocycles
Rigid meromorphic cocycles can be viewed as p-adic analogues of classical modular functions, such as the j-function. Their values at special points are conjectured to be algebraic and highly structured. In this talk, I will discuss the case of rigid meromorphic cocycles for orthogonal groups, with a focus on quadratic spaces of dimension 4. I will highlight the similarities with the theory of Borcherds products on Hilbert modular surfaces. This is an account of joint work with Henri Darmon and Michael Lipnowski, and with Xavier Guitart and Marc Masdeu.
Madhavan Venkatesh, Counting points on varieties over finite fields
This talk is on computing the point counts of algebraic varieties, i.e., number of solutions of a system of polynomial equations over finite fields. The zeta function encodes the point counts over an infinite tower of finite field extensions and enjoys the property of being a rational function. Further, the zeta function can be recovered from certain invariants of the variety in question, using an appropriate cohomology theory. I will review the state of the art on efficient algorithms to compute the zeta function of varieties, including the dimension one case of curves and report on our work (joint with N. Saxena) on surfaces, which involves computing etale cohomology groups with Galois action, addressing a conjecture of Couveignes and Edixhoven
Alexandre Maksoud, Families of p-adic modular forms and their geometry
Since the works of Serre, Hida, and others, modular forms have been known to vary in rich p-adic families. Understanding these families and the Galois representations they encode has been central to major developments such as BSD-type formulas and modularity theorems. This talk, intended as a friendly introduction to the theory of p-adic families, will focus on the local geometry of eigenvarieties near classical weight one forms in the last remaining difficult case. Our result identifies the geometric structure in this setting under a natural non-vanishing condition on certain p-adic regulators, a condition predicted by conjectures in p-adic transcendental number theory. This is joint work with Adel Betina and Alice Pozzi.
Marusia Rebolledo, Beading workshop
We will talk about necklaces (on elliptic curves). A joint work with Christian Wuthrich.