Galois module structure and Jacobians of Fermat curves (original) (raw)

Homomorphisms of abelian varieties

2005

We study Galois properties of points of prime order on an abelian variety that imply the simplicity of its endomorphism algebra. Applications of these properties to hyperelliptic jacobians are discussed. Résumé (Homomorphismes des variétés abéliennes).-Nousétudions les propriétés galoisiennes des points d'ordre fini des variétés abéliennes qui impliquent la simplicité de leur algèbre d'endomorphismes. Nous discutons ceux-ci par rapport aux jacobiennes hyperelliptiques.

A remark on the decomposition of the Jacobian variety of Fermat curves of prime degree

Archiv der Mathematik, 2015

Recently, Barraza-Rojas have described the action of the full automorphisms group on the Fermat curve of degree p, for p a prime integer, and obtained the group algebra decomposition of the corresponding Jacobian variety. In this short note we observe that the factors in such a decomposition are given by the Jacobian varieties of certain p-gonal curves.

Rational Points On Certain Abelian Varieties Over Function Fields

Journal of Number Theory, 1995

We prove a structure theorem on the Mordell-Weil group of Abelian varieties over function fields that arise as twists of Abelian varieties by cyclic covers of quasi projective varieties, in terms of Prym varieties associated to the cyclic covers. Given integers 2 ≤ s ≤ r and a polynomial f (x) of degree r with coefficients in a global field, we apply our result to the twists of Jacobian varieties of super-elliptic curves defined by affine equation y s = f (x) with cyclic covers of certain varieties to get super-elliptic Jacobians with large Mordell-Weil ranks.

Abelian varieties over finitely generated fields and the conjecture of Geyer and Jarden on torsion

Mathematische Nachrichten, 2013

In this paper we prove the Geyer-Jarden conjecture on the torsion part of the Mordell-Weil group for a large class of abelian varieties defined over finitely generated fields of arbitrary characteristic. The class consists of all abelian varieties with big monodromy, i.e., such that the image of Galois representation on ℓ-torsion points, for almost all primes ℓ, contains the full symplectic group.

Abelian varieties over fields of finite characteristic

Central European Journal of Mathematics, 2014

The aim of this paper is to extend our previous results about Galois action on the torsion points of abelian varieties to the case of (finitely generated) fields of characteristic 2.