Le groupe de Selmer des isog\'enies de hauteur un

TL;DR

This paper demonstrates that the Selmer group of a height-one isogeny between abelian varieties over a function field can be embedded into the homomorphism group of certain natural vector bundles on the base variety, revealing a geometric structure.

Contribution

It establishes a new connection between Selmer groups and vector bundle homomorphisms for height-one isogenies over function fields in characteristic p.

Findings

Selmer group embeds into vector bundle homomorphisms

Provides geometric interpretation of Selmer groups

Applicable to abelian varieties over function fields

Abstract

On montre que le groupe de Selmer d'une isog\'enie de hauteur un entre deux vari\'et\'es ab\'eliennes d\'efinies sur le corps de fonctions d'une vari\'et\'e quasi-projective et lisse $V$ sur un corps parfait $k_0$ de caract\'eristique $p>0$ peut \^etre plong\'e dans le groupe des homomorphismes entre deux fibr\'es vectoriels naturels sur $V$. / We show that the Selmer group of an isogeny of height one between two abelian varieties defined on the function field of a smooth and quasi-projective variety $V$ over a perfect field $k_0$ of characteristic $p>0$ can be embedded in the group of homomorphisms between two natural vector bundles on $V$.