Abelian varieties with isogenous reductions

Khare, Chandrashekhar B.; Larsen, Michael

doi:10.5802/crmath.129

Théorie des nombres

Abelian varieties with isogenous reductions

Khare, Chandrashekhar B.¹ ; Larsen, Michael ²

¹ UCLA Department of Mathematics, Box 951555, Los Angeles, CA 90095, USA
² Department of Mathematics, Indiana University, Bloomington, IN 47405, USA

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

Résumé
Abstract

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 $\bar{K}$ . Nous donnons également une majoration du nombre du composantes d’un sous-groupe réductif de ${GL}_{n}$ dont l’intersection avec l’union des classes de conjugaison $ℚ$ -rationnelles de ${GL}_{n}$ 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 $\bar{K}$ . Along the way, we give an upper bound for the number of components of a reductive subgroup of ${GL}_{n}$ whose intersection with the union of $ℚ$ -rational conjugacy classes of ${GL}_{n}$ is Zariski-dense. This can be regarded as a generalization of the Minkowski–Schur theorem on faithful representations of finite groups with rational characters.

Reçu le : 2020-09-07
Révisé le : 2020-10-06
Accepté le : 2020-10-06
Publié le : 2021-01-05

DOI : 10.5802/crmath.129

Affiliations des auteurs :

Khare, Chandrashekhar B. ¹ ; Larsen, Michael ²

¹ UCLA Department of Mathematics, Box 951555, Los Angeles, CA 90095, USA
² Department of Mathematics, Indiana University, Bloomington, IN 47405, USA

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     title = {Abelian varieties with isogenous reductions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1085--1089},
     publisher = {Acad\'emie des sciences, Paris},
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     year = {2020},
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     url = {https://www.numdam.org/articles/10.5802/crmath.129/}
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Khare, Chandrashekhar B.; Larsen, Michael. Abelian varieties with isogenous reductions. Comptes Rendus. Mathématique, Tome 358 (2020) no. 9-10, pp. 1085-1089. doi : 10.5802/crmath.129. https://www.numdam.org/articles/10.5802/crmath.129/

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