  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.
