Erratum: "On the number of prime divisors of the order of elliptic curves modulo p" (Acta Arith. 117 (2005), 341-352)
Erratum: Rational points of bounded height on Fano varieties. Invent. Math. 95, 421-435 (1989).
Espaces homogènes principaux, unités elliptiques et fonctions
Nous étudions la structure de certains espaces homogènes principaux associés aux éléments du groupe de Selmer d’une courbe elliptique à multiplication complexe. Nous utilisons des résultats de Rubin pour construire, à partir des unités elliptiques, des espaces homogènes principaux de structure galoisienne non triviale. Cette construction fournit un lien nouveau entre un problème de structure galoisienne et certaines fonctions -adiques.
Estimates for complete multiple exponential sums
Estimating heights using auxiliary functions
Estimation effective des points entiers d'une famille de courbes algébriques
Étale cohomology and reduction of abelian varieties
In this paper we study the étale cohomology groups associated to abelian varieties. We obtain necessary and sufficient conditions for an abelian variety to have semistable reduction (or purely additive reduction which becomes semistable over a quadratic extension) in terms of the action of the absolute inertia group on the étale cohomology groups with finite coefficients.
Étude de pour des fonctions relatives à et associées à des caractères de degré
Carlitz a défini pour les corps de fonctions l’analogue du réel et Goss l’analogue des fonctions de Dirichlet. Nous prouvons dans un cas particulier qu’il existe des valeurs entières et des caractères pour lesquels peut être rationnel, algébrique ou bien transcendant.
Étude de la courbe elliptique et monogénéité de certains anneaux d’entiers
Étude d'un idéal particulier, d'indice fini dans le carré de l'idéal d'augmentation, associé à un caractère de Dirichlet d'un groupe fini
We describe here two sets of generators of an ideal , of finite index inside the square of the augmentation ideal of , associated to the Dirichlet character of the finite group . That peculiar ideal first appeared in questions related to the computation of class number formulas for abelian non ramified extensions of -fields cf. [2] and [3], satisfying certain special conditions which are outlined in the introduction of [1]. A rough idea of these formulas is given in §§2 and 6.
Euclidean algorithm and Kummer covers with many points.
Euler system for Galois deformations
In this paper, we develop the Euler system theory for Galois deformations. By applying this theory to the Beilinson-Kato Euler system for Hida’s nearly ordinary modular deformations, we prove one of the inequalities predicted by the two-variable Iwasawa main conjecture. Our method of the proof of the Euler system theory is based on non-arithmetic specializations. This gives a new simplified proof of the inequality between the characteristic ideal of the Selmer group of a Galois deformation and the...
Euler systems obtained from congruences between Hilbert modular forms
Euler's concordant forms
Every ordinary symplectic isogeny class in positive characteristic is dense in the moduli.
Existence of nonelliptic mod Galois representations for every .
Existence of rational points on smooth projective varieties
Experimental study of the rank of families of elliptic curves over . (Étude expérimentale du rang de familles de courbes elliptiques sur .)
Explicit 4-descents on an elliptic curve
Explicit bounds for split reductions of simple abelian varieties
Let be an absolutely simple abelian variety over a number field; we study whether the reductions tend to be simple, too. We show that if is a definite quaternion algebra, then the reduction is geometrically isogenous to the self-product of an absolutely simple abelian variety for in a set of positive density, while if is of Mumford type, then is simple for almost all . For a large class of abelian varieties with commutative absolute endomorphism ring, we give an explicit upper bound...