On the Number of Rational Points of Jacobians over Finite Fields

In this article we prove lower and upper bounds for class numbers of algebraic curves defined over finite fields. These bounds turn out to be better than most of the previously known bounds obtained using combinatorics. The methods used in the proof are essentially those from the explicit asymptotic theory of global fields. We thus provide a concrete application of effective results from the asymptotic theory of global fields and their zeta-functions

Generalised Mertens and Brauer-Siegel Theorems

In this article, we prove a generalisation of the Mertens theorem for prime numbers to number fields and algebraic varieties over finite fields, paying attention to the genus of the field (or the Betti numbers of the variety), in order to make it tend to infinity and thus to point out the link between it and the famous Brauer-Siegel theorem. Using this we deduce an explicit version of the generalised Brauer-Siegel theorem under GRH, and a unified proof of this theorem for asymptotically exact families of almost normal number fields

Sur le th\'eor\`eme de Brauer--Siegel g\'en\'eralis\'e

We study a conjecture of Tsfasman and Vladuts which posits a general version of the Brauer--Siegel theorem for any asymptotically exact family of number fields. We suggest an approach which, not only allows to unify the proofs of several previous results towards this conjecture as well as generalise these to a relative setting, but also yields new unconditional cases of the conjecture. We exhibit new sets of conditions which ensure that a family of number fields unconditionally satisfies the conjecture of Tsfasman and Vladuts. We thus prove that this conjecture holds for any asymptotically good family of number fields contained in the solvable closure of a given number field. We further give a number of explicit examples of such families, such as that of an infinite global field contained in a $p$-class field tower.Comment: In French. 22 pages. V2 improves the exposition and strenghtens some of the statements. Comments still welcome

On M-functions associated with modular forms

International audienceLet $f$ be a primitive cusp form of weight $k$ and level $N,$ let $\chi$ be a Dirichlet character of conductor coprime with $N,$ and let $\mathfrak{L}(f\otimes \chi, s)$ denote either $\log L(f\otimes \chi, s)$ or $(L'/L)(f\otimes \chi, s).$ In this article we study the distribution of the values of $\mathfrak{L}$ when either $\chi$ or $f$ vary. First, for a quasi-character $\psi\colon \mathbb{C} \to \mathbb{C}^\times$ we find the limit for the average $\mathrm{Avg}_\chi \psi(L(f\otimes\chi, s)),$ when $f$ is fixed and $\chi$ varies through the set of characters with prime conductor that tends to infinity. Second, we prove an equidistribution result for the values of $\mathfrak{L}(f\otimes \chi,s)$ by establishing analytic properties of the above limit function. Third, we study the limit of the harmonic average $\mathrm{Avg}^h_f \psi(L(f, s)),$ when $f$ runs through the set of primitive cusp forms of given weight $k$ and level $N\to \infty.$ Most of the results are obtained conditionally on the Generalized Riemann Hypothesis for $L(f\otimes\chi, s).