Number Theory Seminar: Exceptional splitting of reductions of abelian surfaces with real multiplication
Seminar | September 11 | 4:10-5 p.m. | 891 Evans Hall | Canceled
Yunqing Tang, Princeton University
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 $\mathfrak p$ for a density one set of primes $\mathfrak p$. One may ask whether its complement, the density zero set of primes $\mathfrak p$ such that the reduction of $A$ modulo $\mathfrak p$ is not geometrically simple, is infinite. Such question is analogous to the study of exceptional mod $\mathfrak p$ 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.