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Classe de conjugaison du frobenius des variétés abéliennes à réduction ordinaire

Rutger Noot (1995)

Annales de l'institut Fourier

Soient X une variété abélienne sur un corps de nombres E et G son groupe de Mumford–Tate. Soit v une valuation de E et pour tout nombre premier ℓ tel que v ( ℓ ) = 0 , soit F ℓ ∈ G ( Q ℓ ) l’automorphisme de Frobenius (géométrique) de la cohomologie étale ℓ -adique de X . On montre que si X a une bonne réduction ordinaire en v , alors il existe F ∈ G ( Q ) tel que, pour tout ℓ , F ℓ soit conjugué à F dans G ( Q ℓ ) . On montre un résultat analogue pour le frobenius de la cohomologie cristalline de la réduction de X modulo v .

Corestriction of central simple algebras and families of Mumford-type

Federica Galluzzi (1999)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let M be a family of Mumford-type, that is, a family of polarized complex abelian fourfolds as introduced by Mumford in [9]. This family is defined starting from a quaternion algebra A over a real cubic number field and imposing a condition to the corestriction of such A . In this paper, under some extra conditions on the algebra A , we make this condition explicit and in this way we are able to describe the polarization and the complex structures of the fibers. Then, we look at the non simple C ⁢ M -fibers...

Degré d’une extension de 𝐐 p nr sur laquelle J 0 ( N ) est semi-stable

Mohamed Krir (1996)

Annales de l'institut Fourier

Soit N un entier ≥ 1 . Pour un nombre premier p on note Q p nr l’extension maximale non ramifiée de Q p . Supposons que p v divise exactement N . Alors, en utilisant les travaux de Carayol et la théorie du corps de classes local, on détermine une extension E v de Q p nr sur laquelle la jacobienne J 0 de la courbe modulaire de X 0 ( N ) admet une réduction semi-stable, puis on donne une estimation de son degré.

Descent via (3,3)-isogeny on Jacobians of genus 2 curves

Nils Bruin, E. Victor Flynn, Damiano Testa (2014)

Acta Arithmetica

We give a parametrization of curves C of genus 2 with a maximal isotropic (ℤ/3)² in J[3], where J is the Jacobian variety of C, and develop the theory required to perform descent via (3,3)-isogeny. We apply this to several examples, where it is shown that non-reducible Jacobians have non-trivial 3-part of the Tate-Shafarevich group.

Diophantine approximation on algebraic varieties

Michael Nakamaye (1999)

Journal de théorie des nombres de Bordeaux

We present an overview of recent advances in diophantine approximation. Beginning with Roth's theorem, we discuss the Mordell conjecture and then pass on to recent higher dimensional results due to Faltings-Wustholz and to Faltings respectively.

Distribution des points de petite hauteur dans les groupes multiplicatifs

Francesco Amoroso, Sinnou David (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We prove a new lower bound for the height of points on a subvariety  V of a multiplicative torus, which lie outside the union of torsion subvarieties of  V . Although lower bounds for the heights of these points where already known (decreasing multi-exponential function of the degree for Scmhidt and Bombieri–Zannier, [Sch], [Bo-Za], and inverse monomial in the degree by the second author of this note and P. Philippon, [Da-Phi]), our method provesup to an ε the sharpest conjectures that can be formulated....

Équidistribution des sous-variétés de petite hauteur

Pascal Autissier (2006)

Journal de Théorie des Nombres de Bordeaux

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.

Étale cohomology and reduction of abelian varieties

A. Silverberg, Yu. G. Zarhin (2001)

Bulletin de la Société Mathématique de France

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.

Explicit bounds for split reductions of simple abelian varieties

Jeffrey D. Achter (2012)

Journal de Théorie des Nombres de Bordeaux

Let X / K be an absolutely simple abelian variety over a number field; we study whether the reductions X 𝔭 tend to be simple, too. We show that if End ( X ) is a definite quaternion algebra, then the reduction X 𝔭 is geometrically isogenous to the self-product of an absolutely simple abelian variety for 𝔭 in a set of positive density, while if X is of Mumford type, then X 𝔭 is simple for almost all 𝔭 . For a large class of abelian varieties with commutative absolute endomorphism ring, we give an explicit upper bound...

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