Non-simple abelian varieties in a family: arithmetic approaches

TL;DR

This paper explores how the property of being geometrically simple in the generic fiber of a family of abelian varieties extends to other fibers, using arithmetic methods and bounds on covers with level structures.

Contribution

It generalizes previous results on geometric simplicity from specific Jacobian families to all families with large geometric monodromy using an arithmetic approach.

Findings

Extended the property of geometric simplicity to all fibers in certain abelian variety families.

Applied Heath-Brown-type bounds to covers with level structures for new results.

Optimized covers to improve the extension of properties across fibers.

Abstract

Inspired by the work of Ellenberg, Elsholtz, Hall, and Kowalski, we investigate how the property of the generic fiber of a one-parameter family of abelian varieties being geometrically simple extends to other fibers. In \cite{EEHK09}, the authors studied a special case involving specific one-parameter families of Jacobians of curves using analytic methods. We generalize their results, particularly Theorem B, to all families of abelian varieties with big geometric monodromy, employing an arithmetic approach. Our method applies Heath-Brown-type bounds on certain covers with level structures and optimizes the covers to derive the desired results.