A Bound for the Image Conductor of a Principally Polarized Abelian Variety with Open Galois Image

Abstract

Let A be a principally polarized abelian variety of dimension g over a number field K. Assume that the image of the adelic Galois representation of A is open. Then there exists a positive integer m so that the Galois image of A is the full preimage of its reduction modulo m. The least m with this property, denoted m A , is called the image conductor of A. A recent paper [2] established an upper bound for m A , in terms of standard invariants of A, in the case that A is an elliptic curve without complex multiplication. In this note, we generalize the aforementioned result to provide an analogous bound in arbitrary dimension.