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

Jacob Mayle

Published 2020-01-20Version 1

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 (arXiv:1904.10431) 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.