A bound for the image conductor of a principally polarized abelian variety with open Galois image

Abstract

Let A A be a principally polarized abelian variety of dimension g g over a number field K K . Assume that the image of the adelic Galois representation of A A is an open subgroup of G S p 2 g ( Z ^ ) GSp_{2g}(\hat {\mathbb {Z}}) . Then there exists a positive integer m m so that the Galois image of A A is the full preimage of its reduction modulo m m . The least m m with this property, denoted m A m_A , is called the image conductor of A A . Jones [Pacific J. Math. 308 (2020), pp. 307–331] recently established an upper bound for m A m_A , in terms of standard invariants of A A , in the case that A A is an elliptic curve without complex multiplication. In this paper, we generalize the aforementioned result to provide an analogous bound in arbitrary dimension.