On the Frobenius fields of abelian varieties over number fields

Burungale, Ashay A.; Hida, Haruzo; Lai, Shilin

doi:10.2140/ant.2026.20.577

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On the Frobenius fields of abelian varieties over number fields

Ashay A. Burungale, Haruzo Hida and Shilin Lai

Vol. 20 (2026), No. 3, 577–601

DOI: 10.2140/ant.2026.20.577

Abstract

Let $A$ be a non-CM simple abelian variety over a number field $K$ . For a place $v$ of $K$ where $A$ has good reduction, let $F (A, v)$ denote the Frobenius field generated by the corresponding Frobenius eigenvalues. If $A$ has connected monodromy groups, we show that the set of places $v$ such that $F (A, v)$ is isomorphic to a fixed number field has upper Dirichlet density zero. Moreover, assuming the GRH, we give a power saving upper bound for the number of such places.

Keywords

abelian varieties, Frobenius fields, Mumford–Tate groups

Mathematical Subject Classification

Primary: 14K15

Milestones

Received: 22 June 2024

Revised: 14 January 2025

Accepted: 3 March 2025

Published: 24 March 2026

Authors

Ashay A. Burungale
Department of Mathematics University of Texas at Austin Austin, TX United States

Haruzo Hida
Department of Mathematics University of California, Los Angeles Los Angeles, CA United States

Shilin Lai
Department of Mathematics University of Michigan Ann Arbor, MI United States

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Vol. 20, No. 3, 2026