Exposés

A remark on special values of Gaussian hypergeometric functions

Xavier Caruso

Résumé :

In this talk, I will share some recent observations about the special value at $1$ of the Gaussian hypergeometric function ${}_2 F_1 (a, b; c; z)$ when the parameters $a$, $b$ and $c$ are rational numbers subject to the condition $a + b - 2c \in \mathbb Z$.

More precisely, we will give meaningful values for ${}_2 F_1 (a, b; c; 1)$ over $\mathbb C$, over $\mathbb \F_p$ and over $\mathbb Q_p$. Although our definitions are very dependant on the base field, we will show that the obtained values are
surprinsingly compatible in a strong sense.

A Linear Independence of the Multiple Zeta Values in Positive Characteristics

Hojin Kim

Résumé :

The Zagier-Hoffman’s conjectures describe the dimension and a basis set for the $\mathbb{Q}$-vector space spanned by classical Multiple Zeta Values (MZV's). The function field analogy allows us to define the MZV's in positive characteristics, and the Zagier-Hoffman’s conjectures have been established for the MZV's in positive characteristics over the function field $\mathbb{F}_q(\theta)$ by Im, Le, Ngo Dac, Pham and the speaker (see also the work of Chang, Chen, and Mishiba). Building on this result, this talk presents ongoing research on the linear independence of the MZVs in positive characteristics over $\mathbb{F}_q$. This is joint work with Bo-Hae Im and Tuan Ngo Dac. 

A P-adic class formula for Drinfeld modules

Alexis Lucas

Résumé :

In this talk, for $L/\F_q(t)$ a finite extension with ring of integer denotes by $\mathscr{O}_L$, I will introduce the $P$-adic $L$-series associated with  Drinfeld $\F_q[t]$-modules defined over $\mathscr{O}_L$. I will then explain a P-adic class formula à la Taelman, linking the $P$-adic $L$-series to a certain $P$-adic regulator and the Taelman class module. I will then give necessary and sufficient conditions for the vanishing of this series under certain conjectures. Finally, if time permits, we will consider the vanishing order at $z=1$ of the $P$-adic $L$-series in the case $L=K$. 

The refined class number formula for Drinfeld modules

Daniel Macias Castillo

Résumé :

In 2012, Taelman proved an analogue of the Analytic Class Number Formula, for the Goss $L$-values that are associated to Drinfeld modules. He also explicitly stated that `it should be possible to formulate and prove an equivariant version' of this formula.

In joint work with Mar´ıa In´es de Frutos Fern´andez and Daniel Mart´ınez Marqu´es, we formulate and prove an equivariant, or ‘refined’, version of Taelman’s formula.

In this talk we will review Taelman's work and discuss the formulation of the refined formula, as well as some additional explicit consequences. If time permits, we will also briefly touch on related work in progress.

Determining t-motives and dual t-motives

Andreas Maurischat

Résumé :

In this talk, we present an algorithm to compute Anderson t-motives and dual t-motives attached to abelian Anderson t-modules. Reversely, for a given object in the category of (dual) t-motives, we show how to decide whether it is the (dual) t-motive associated to a t-module, and determine that t-module.

As it turned out, all these algorithms origin from a single algorithm about modules over skew polynomial rings. Hence most of the talk, doesn't deal with Anderson t-modules at all.

Exemples de formes automorphes en caractéristique positive

Hassan Oukhaba

Résumé :

L'idée est d'expliquer le cheminement qui permet de passer de l'espace des fonctions analytiques rigides sur une courbe de Mumford, obtenue à partir d'une algèbre de quaternion sur une corps de fonctions, aux formes automorphes associées au groupe des inversibles de l'algèbre en question.

Functional identities for zeta functions over curves

Federico Pellarin

Résumé :

Riemann’s zeta function, a complex valued function, interpolates zeta values at integers greater than two and thanks to its functional equation, we can deduce Euler’s famous identities, first discovered in 1735. 

In the years 1980 Goss introduced zeta and L-values interpolating certain zeta values introduced by Carlitz in 1935 that are elements of complete algebraically closed fields of positive characteristic. A functional equation for these functions is not known.

The aim of this talk is to discuss Goss philosophy and compare it with a novel approach, where analytic functions interpolating Carlitz zeta values (and more) are introduced. We will conclude describing the results of Ferraro (2022) where a functional identity is proved for certain analytic functions over any smooth projective curve over a finite field.

Perrin-Riou's main conjecture for modular forms

Maria Rosaria Pati

Résumé :

In this talk, I will state and sketch a proof of the counterpart for a higher (even) weight newform f of Perrin-Riou’s Heegner point main conjecture for elliptic curves ("Heegner cycle main conjecture" for f). Our strategy of proof builds on ideas of Bertolini and Darmon, as elaborated by Howard, and consists in the construction of a so-called bipartite Euler system, that is a collection of cohomology classes and p-adic L-functions attached to congruent modular forms. This is joint work with Matteo Longo and Stefano Vigni.

Algebraic independence of E and G-functions

Daniel Vargas Montoya

Résumé :

The E and G-functions are series introduced by Siegel in 1929  with the aim of generalizing the Lindemann–Weierstrass and
Hermite–Lindemann theorems. Since then, the algebraic independence of these series, as well as of their values at algebraic points, has been the subject of numerous studies. In this talk, we present a method based on reduction modulo p and differential Galois theory to study the algebraic independence of certain E- and G-functions.

Aspects p-adiques des formes de Maass

Jan Vonk

Résumé :

Suivant une suggestion de Blasius et Calegari, on explore quelques analogies entre les formes d'onde de Maass et les formes propres surconvergentes (p-adiques), en nous concentrant sur leurs ensembles de zéros. Ce travail est une collaboration avec Paolo Bordignon.