Abelian varieties with isogenous reductions

Khare, Chandrashekhar B.; Larsen, Michael

Comptes Rendus. Mathématique, Tome 358 (2020) no. 9-10, pp. 1085-1089.

Soient $A_1$ et $A_2$ deux variétés abéliennes sur un corps de nombres $K$. Nous montrons que, s’il existe un morphisme non trivial de variétés abéliennes entre réductions de $A_1$ et $A_2$ pour une proportion suffisamment grande d’idéaux premiers, il existe un morphisme non trivial $A_1\to A_2$ sur $\overline{K}$. Nous donnons également une majoration du nombre du composantes d’un sous-groupe réductif de ${\mathrm{GL}}_n$ dont l’intersection avec l’union des classes de conjugaison $\mathbb{Q}$-rationnelles de ${\mathrm{GL}}_n$ est dense pour la topologie de Zariski ; c’est une généralisation d’un théorème de Minkowski–Schur sur les représentations fidèles des groupes finis à caractère rationnel.

Let $A_1$ and $A_2$ be abelian varieties over a number field $K$. We prove that if there exists a non-trivial morphism of abelian varieties between reductions of $A_1$ and $A_2$ at a sufficiently high percentage of primes, then there exists a non-trivial morphism $A_1\to A_2$ over $\overline{K}$. Along the way, we give an upper bound for the number of components of a reductive subgroup of ${\mathrm{GL}}_n$ whose intersection with the union of $\mathbb{Q}$-rational conjugacy classes of ${\mathrm{GL}}_n$ is Zariski-dense. This can be regarded as a generalization of the Minkowski–Schur theorem on faithful representations of finite groups with rational characters.

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