Nouveaux théorèmes d’isogénie

Abstract

— Given a finitely generated field extension K of the rational numbers and an abelian variety C over K, we consider the class of all abelian varieties over K which are isogenous (over K) to an abelian subvariety of a power of C. We show that there is a single, naturally constructed abelian variety C in the class whose ring of endomorphisms controls all isogenies in the class. Precisely, this means that if d is the discriminant of this ring then for any pair of isogenous abelian varieties in the class there exists an isogeny between them whose kernel has exponent at most d. Furthermore we prove that, for any element A in the class, the same number d governs several invariants attached to A such as the smallest degree of a polarisation on A, the discriminant of its ring of endomorphisms or the size of the invariant part of its geometric Brauer group. All these are bounded only in terms of d and the dimension of A. In the case where K is a number field we can go further and show that the period theorem applies to C in a natural way and gives an explicit bound for d in terms of the degree of K, the dimension of C and the stable Faltings height of C. This in turn yields explicit upper bounds for all the previous quantities related to isogenies, polarisations, endomorphisms, Brauer groups which significantly improve known results. MSC 2020 : 11G10 (14K02, 11R54, 14K15). Mots-clefs : variété abélienne, isogénie, ordre maximal, ordre principal, anneau de Lefschetz, module de Tate, polarisation, groupe de Brauer, réseau des périodes, théorème des périodes.