# Exceptional splitting of reductions of abelian surfaces

## TL;DR

This paper proves that abelian surfaces with real multiplication have infinitely many primes of split reduction, using Arakelov intersection theory on Hilbert modular surfaces, extending results known for elliptic curves.

## Contribution

It establishes the infinitude of split reduction primes for abelian surfaces with real multiplication, a significant extension of previous elliptic curve results.

## Key findings

- Infinitely many split reduction primes for abelian surfaces with real multiplication
- Density-zero set of primes pertaining to reduction is infinite
- Utilizes Arakelov intersection theory on Hilbert modular surfaces

## Abstract

Heuristics based on the Sato--Tate conjecture suggest that an abelian surface defined over a number field has infinitely many places of split reduction. We prove this result for abelian surfaces having real multiplication. Similar to Charles' theorem on exceptional isogeny of reductions of a given pair of elliptic curves and Elkies' theorem on supersingular reductions of a given elliptic curve, our theorem shows that a density-zero set of primes pertaining to the reduction of abelian varieties is infinite. The proof relies on the Arakelov intersection theory on Hilbert modular surfaces.