Diophantische Analysis und Modulfunktionen.
Dirichlet motives via modular curves
Drinfeld shtukas and applications. (Chtoucas de Drinfeld et applications.)
Endomorphism algebras of motives attached to elliptic modular forms
We study the endomorphism algebra of the motive attached to a non-CM elliptic modular cusp form. We prove that this algebra has a sub-algebra isomorphic to a certain crossed product algebra . The Tate conjecture predicts that is the full endomorphism algebra of the motive. We also investigate the Brauer class of . For example we show that if the nebentypus is real and is a prime that does not divide the level, then the local behaviour of at a place lying above is essentially determined...
Equations of hyperelliptic modular curves
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.
Facteurs -simples de de grande dimension et de grand rang
Fake CM and the stable model of .
Familles de polynômes liées aux courbes modulaires X₁(l) unicursales et points rationnels non-triviaux de courbes elliptiques quotient
Fields of definition of building blocks with quaternionic multiplication
Fields of definition of -curves
Let be a -curve with no complex multiplication. In this note we characterize the number fields such that there is a curve isogenous to having all the isogenies between its Galois conjugates defined over , and also the curves isogenous to defined over a number field such that the abelian variety Res obtained by restriction of scalars is a product of abelian varieties of GL-type.
Fonctions hypergéométriques en plusieurs variables et espaces des modules de variétés abéliennes
Fundamental domains for Shimura curves
We describe a process for defining and computing a fundamental domain in the upper half plane of a Shimura curve associated with an order in a quaternion algebra . A fundamental domain for realizes a finite presentation of the quaternion unit group, modulo units of its center. We give explicit examples of domains for the curves . 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....
Galois orbits and equidistribution: Manin-Mumford and André-Oort.
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.
Galois theory and torsion points on curves
In this paper, we survey some Galois-theoretic techniques for studying torsion points on curves. In particular, we give new proofs of some results of A. Tamagawa and the present authors for studying torsion points on curves with “ordinary good” or “ordinary semistable” reduction at a given prime. We also give new proofs of : (1) the Manin-Mumford conjecture : there are only finitely many torsion points lying on a curve of genus at least embedded in its jacobian by an Albanese map; and (2) the...
Genera of arithmetic Fuchsian groups
Generalizing the Birch-Stephens theorem. I. Modular curves.
Generators and defining equation of the modular function field of the group Γ₁(N)
Growth of Selmer groups of Hilbert modular forms over ring class fields
We prove non-trivial lower bounds for the growth of ranks of Selmer groups of Hilbert modular forms over ring class fields and over certain Kummer extensions, by establishing first a suitable parity result.
Hauteur des correspondances de Hecke
L’objectif de cet article est de mesurer la complexité arithmétique de la courbe modulaire en fonction du niveau . Pour ce faire, on utilise un morphisme fini (de degré 1 sur son image) de vers une variété fixe et on calcule la hauteur au sens d’Arakelov de l’image de ce morphisme. La hauteur employée est directement reliée à la hauteur de Faltings des courbes elliptiques. On a besoin pour cela de considérer une théorie d’Arakelov pour les faisceaux inversibles hermitiens -singuliers (au...
Hecke action and the degree of the modular parameterization