Exceptional splitting of reductions of abelian surfaces

Abstract

Heuristics based on the Sato–Tate conjecture and the Lang–Trotter philosophy suggest that an abelian surface defined over a number field has infinitely many places of split reduction. We prove this result for abelian surfaces with real multiplication. As in previous work by Charles and Elkies, this shows that a density $0$ set of primes pertaining to the reduction of abelian varieties is infinite. The proof relies on the Arakelov intersection theory on Hilbert modular surfaces.