Endlichkeitssätze für abelsche Varietäten über Zahlkörpern.
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 for Mahler measure and isogenies
We study some functional equations between Mahler measures of genus-one curves in terms of isogenies between the curves. These equations have the potential to establish relationships between Mahler measure and especial values of -functions. These notes are based on a talk that the author gave at the “Cuartas Jornadas de Teoría de Números”, Bilbao, 2011.
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.
Equidistribution and the heights of totally real and totally p-adic numbers
C. J. Smyth was among the first to study the spectrum of the Weil height in the field of all totally real numbers, establishing both lower and upper bounds for the limit infimum of the height of all totally real integers, and determining isolated values of the height. Later, Bombieri and Zannier established similar results for totally p-adic numbers and, inspired by work of Ullmo and Zhang, termed this the Bogomolov property. In this paper, we use results on equidistribution of points of low height...
Équidistribution des sous-variétés de petite hauteur
On montre dans cet article que le théorème d’équidistribution de Szpiro-Ullmo-Zhang concernant les suites de petits points sur les variétés abéliennes s’étend au cas des suites de sous-variétés. On donne également une version quantitative de ce résultat.
Equidistribution of dynamically small subvarieties over the function field of a curve
Equidistribution of Small Points, Rational Dynamics, and Potential Theory
Given a rational function on of degree at least 2 with coefficients in a number field , we show that for each place of , there is a unique probability measure on the Berkovich space such that if is a sequence of points in whose -canonical heights tend to zero, then the ’s and their -conjugates are equidistributed with respect to .The proof uses a polynomial lift of to construct a two-variable Arakelov-Green’s function for each . The measure is obtained by taking the...
Equidistribution of small subvarieties of an Abelian variety.
Equivalences between elliptic curves and real quadratic congruence function fields
In 1994, the well-known Diffie-Hellman key exchange protocol was for the first time implemented in a non-group based setting. Here, the underlying key space was the set of reduced principal ideals of a real quadratic number field. This set does not possess a group structure, but instead exhibits a so-called infrastructure. More recently, the scheme was extended to real quadratic congruence function fields, whose set of reduced principal ideals has a similar infrastructure. As always, the security...
Équivalences rationnelle et numérique sur certaines variétés de type abélien sur un corps fini
Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves.
Errata-Corrige : “The Mordell conjecture revisited”
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.