We compute, in a unified way, the equations of all hyperelliptic modular curves. The main tool is provided by a class of modular functions introduced by Newman in 1957. The method uses the action of the hyperelliptic involution on the cusps.

Let $C$ be a $\mathbb{Q}$-curve with no complex multiplication. In this note we characterize the number fields $K$ such that there is a curve $C^{\text{'}}$ isogenous to $C$ having all the isogenies between its Galois conjugates defined over $K$, and also the curves $C^{\text{'}}$ isogenous to $C$ defined over a number field $K$ such that the abelian variety Res${}_{K/\mathbb{Q}}(C^{\text{'}}/K)$ obtained by restriction of scalars is a product of abelian varieties of GL${}_2$-type.

Ce papier présente les récents progrès concernant les fonctions zêta des hauteurs associées à la conjecture de Manin. En particulier, des exemples où on peut prouver un prolongement méromorphe de ces fonctions sont détaillés.

On montre, sous certaines hypothèses un résultat en direction de la conjecture de Serre pour $GS{p}_4$ formulée dans un autre article avec F. Herzig : si la représentation résiduelle associée à une forme de Siegel de genre $2$, de niveau premier à $p$, $p$-ordinaire de poids $p$-petit, laisse stables deux droites (au lieu d’une) dans un plan lagrangien, alors cette forme possède une forme compagnon de poids prescrit. Notre méthode consiste à traduire, grâce au théorème de comparaison mod. $p$ de Faltings, l’existence...

We describe a process for defining and computing a fundamental domain in the upper half plane $\mathscr{H}$ of a Shimura curve $X_0^D\left(N\right)$ associated with an order in a quaternion algebra $A/𝐐$. A fundamental domain for $X_0^D\left(N\right)$ realizes a finite presentation of the quaternion unit group, modulo units of its center. We give explicit examples of domains for the curves $X_0^6\left(1\right),X_0^{15}\left(1\right),\text{and}{X}_0^{35}\left(1\right)$. The first example is a classical example of a triangle group and the second is a corrected version of that appearing in the book of Vignéras [13], due to Michon....

We overview a unified approach to the André-Oort and Manin-Mumford conjectures based on a combination of Galois-theoretic and ergodic techniques. This paper is based on recent work of Klingler, Ullmo and Yafaev on the André-Oort conjecture, and of Ratazzi and Ullmo on the Manin-Mumford conjecture.

We describe the rigid geometry of the first layer in the tower of coverings of the $p$-adic upper half plane constructed by Drinfeld. Using our results, we describe the stable fiber at p of certain Shimura curves.

Let $f$ be a Hecke–Maass cusp form of eigenvalue $\lambda$ and square-free level $N$. Normalize the hyperbolic measure such that $\mathrm{vol}\left(Y_0\left(N\right)\right)=1$ and the form $f$ such that ${\parallel f\parallel }_2=1$. It is shown that ${\parallel f\parallel }_{\infty }{\ll }_{ϵ}{\lambda }^{\frac{5}{24}+ϵ}{N}^{\frac{1}{3}+ϵ}$ for all $ϵ>0$. This generalizes simultaneously the current best bounds in the eigenvalue and level aspects.

The aim of these notes is to provide an introduction to the subject of integral canonical models of Shimura varieties, and then to sketch a proof of the existence of such models for Shimura varieties of Hodge and, more generally, abelian type. For full details the reader is refered to [Ki 3].