Exceptional splitting of reductions of abelian surfaces with real multiplication

Workshop on Motives, Galois Representations and Cohomology Around the Langlands Program
Topic:Exceptional splitting of reductions of abelian surfaces with real multiplication
Speaker:Yunqing Tang
Affiliation:Princeton University
Date:Thursday, November 9
Time/Room:2:30pm - 3:30pm/S-101
Video Link:https://video.ias.edu/MotivesGaloisRepsandCohomology/2017/1109-YunqingTang

Abstract: Zywina showed that after passing to a suitable field extnesion, every abelian surface $A$ with real multiplication over some number field has geometrically simple reduction modulo $\frak{p}$ for a density one set of primes $\frak{p}$. One may ask whether its complement, the density zero set of primes $\frak{p}$ such that the reduction of $A$ modulo $\frak{p}$ is not geometrically simple, is infinite. Such question is analogous to the study of exceptional mod $\frak{p}$ 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.