On the Frobenius fields of abelian varieties over number fields

TL;DR

The paper investigates Frobenius fields F(A,v) attached to a non-CM simple abelian variety A over a number field K at places of good reduction. By leveraging a compatible system of Galois representations and Frobenius tori within the algebraic monodromy group, it proves that the set of places v for which F(A,v) is isomorphic to a fixed field M has upper Dirichlet density 0 under the connected monodromy hypothesis, and, under GRH, provides a power-saving bound for their count depending on the Mumford–Tate data. The argument combines finite reductive-group volume bounds, effective Chebotarev, and the Selberg sieve, with a refinement via central isogenies and bounding sets to achieve uniform control across p. In the generic case where the semisimple quotient is PGSp_{2g}, explicit exponents are obtained, illustrating the strength of the approach across dimensions. The methods offer a uniform framework applicable to compatible systems beyond abelian varieties and suggest extensions to function fields and broader Galois-representation contexts.

Abstract

Let $A$ be a non-CM simple abelian variety over a number field $K$. For a place $v$ of $K$ such that $A$ has good reduction at $v$, let $F(A,v)$ denote the Frobenius field generated by the corresponding Frobenius eigenvalues. Assuming $A$ 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. Assuming the GRH, we give a power saving upper bound for the number of such places.

Theorems & Definitions (59)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Corollary 1.4
• Remark 1.5
• Theorem 2.1
• Remark 2.3
• Theorem 2.4
• Theorem 2.5
• proof