# Reductions of abelian surfaces over global function fields

## TL;DR

This paper proves that certain non-isotrivial ordinary abelian surfaces over global function fields are infinitely often isogenous to a product of elliptic curves at various places, under specific conditions on real multiplication.

## Contribution

It establishes the existence of infinitely many places where the abelian surface splits into elliptic curves, extending understanding of abelian surface reductions over function fields.

## Key findings

- Infinitely many places with isogeny to elliptic curve products
- Conditions on real multiplication are crucial
- Advances knowledge of abelian surface behavior over function fields

## Abstract

Let $A$ be a non-isotrivial ordinary abelian surface over a global function field with good reduction everywhere. Suppose that $A$ does not have real multiplication by any real quadratic field with discriminant a multiple of $p$. We prove that there are infinitely many places modulo which $A$ is isogenous to the product of two elliptic curves.