Mathematics

Zywina showed that after passing to a suitable field extension, every abelian surface A with real multiplication over some number field has geometrically simple reduction modulo p for a density one set of primes p. One may ask whether its complement, the density zero set of primes p such that the reduction of A modulo p is not geometrically simple, is infinite. Such question is analogous to the study of exceptional mod p isogeny between two elliptic curves in the recent work of Charles. In this talk, I will show that abelian surfaces over number fields with real multiplication have infinitely many non-geometrically-simple reductions. This is joint work with Ananth Shankar.