Abelian varieties with isogenous reductions

Chandrashekhar B. Khare; Michael Larsen

doi:10.5802/crmath.129

Number Theory

Abelian varieties with isogenous reductions

Chandrashekhar B. Khare ¹ ; Michael Larsen 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, Volume 358 (2020) no. 9-10, pp. 1085-1089.

Abstract
Résumé

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.

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.

Received: 2020-09-07
Revised: 2020-10-06
Accepted: 2020-10-06
Published online: 2021-01-05

DOI: 10.5802/crmath.129

Author's affiliations:

Chandrashekhar B. Khare ¹; Michael Larsen ²

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Chandrashekhar B. Khare and Michael Larsen},
     title = {Abelian varieties with isogenous reductions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1085--1089},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {9-10},
     year = {2020},
     doi = {10.5802/crmath.129},
     language = {en},
}

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Chandrashekhar B. Khare; Michael Larsen. Abelian varieties with isogenous reductions. Comptes Rendus. Mathématique, Volume 358 (2020) no. 9-10, pp. 1085-1089. doi : 10.5802/crmath.129. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.129/

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