We give explicit upper and lower bounds on the size of the coefficients of the modular polynomials $\Phi_N$ for the $j$-invariant. These bounds make explicit the best previously known asymptotic bounds. This is joint work with Fabien Pazuki and Florian Breuer.

Exposé du rencontre OAK2 (Observables – Actions – Quantification)

On introduit une version quantitative de la cohomologie polynomiale des groupes et on montre qu’elle coïncide avec la cohomologie de groupes classique sous des hypothèses de remplissage polynomial. Dans cet exposé on présentera une ou deux applications de ce résultat, en fonction du temps.
1) On montre que les nombres de Betti des groupes nilpotents sont invariants par équivalence mesurable L^p, à partir d’un p assez grand explicite.
2) On obtient des nouvelles annulations de cohomologie unitaire et banachique pour des réseaux non uniformes dans des groupes de Lie simples de rang 1.

In 1996 Gilles Carron introduced the notion of nonparabolic operators at infinity and showed that such operators are Fredholm in the usual sense. The question that we could ask ourselves is: can we extend the notion of Carron to $\Gamma$-operators on Galois coverings and still have a notion of Fredholmness upstairs ?

In this talk we will first introduce the notion of an $\mathcal{N}\Gamma$-Hilbert space as introduced by Wolfgang Lück in 1997. Then, we will introduce the notion of $\Gamma$-Fredholm operators defined between $\mathcal{N}\Gamma$-Hilbert spaces. And finally, the goal of the presentation is to define nonparabolic $\Gamma$-near infinity operators and show how they induce $\Gamma$-Fredholmness on admissible domains.

Many arguments in mathematics rely on the introduction of an auxiliary function that is compactly supported. Sometimes this compactly supported function is required to satisfy some additional regularity, but this cannot be pushed too far. It can for instance not be analytic as follows by the identity principle.

In this talk, we wish to present a technique that may bypass this obstruction. Namely, instead of considering a single function, we shall construct a sequence of function that are supported in the same compact, and satisfy some good uniform bounds on their derivatives.

This method is a powerful tool as it can, in some circumstances, turn a heuristic argument relying on a compactly supported analytic auxiliary function, into a rigorous proof. As an application, we shall discuss how this technique leads to improvements in some quantified Tauberian theorems.

La suite d’Oldenburger-Kolakoski est l’unique mot infini sur l’alphabet {1,2} qui commence par un “1” et est point fixe de l’opérateur de dérivation. En 1991, M.S. Keane conjecture que cette suite admet une fréquence d’1/2 pour la lettre “1”.

Les suites dites “auto-descriptives” sont une généralisation du mot d’Oldenburger-Kolakoski. Ces suites sont en bijection naturelle avec l’ensemble de toutes les suites sur l’alphabet {1,2} : une suite auto-descriptive est dite “dirigée” par son homologue naturelle sur {1,2}. Est-il possible d’inférer les fréquences de lettres de l’une à partir de l’autre ?

Je présenterai dans cet exposé deux approches à cette question : l’une probabiliste (Boisson, Jamet, Marcovici — 2024), l’autre analytique (Akiyama, Jamet, Marcovici, T.C. — 2024).

One of the central topics in number theory is the study of $L$-functions and the distribution of their zeros. For example, the celebrated Prime Number Theorem is equivalent to the fact that the Riemann zeta function $\zeta(s)$ does not vanish on the line $\text{Re}(s)=1$. In this talk, we will focus on quadratic Dirichlet $L$-functions: in particular, the real zeros of the derivative of quadratic Dirichlet $L$-functions $L^\prime (s, \chi_d)$, where $d$ ranges over fundamental discriminants. Baker and Montgomery conjectured that there are $\asymp \log \log |d|$ real zeros of $L^\prime(s, \chi_d)$ in the interval $[1/2, 1]$ for almost all fundamental discriminants $d$. We will highlight some recent exciting progress that comes close to proving this conjecture and then outline the proof, which is based on ideas coming from analytic and probabilistic number theory. This is based on recent joint work with Youness Lamzouri.