Arithmetic Geometry and Number Theory RTG Seminar: Exceptional splitting of reductions of abelian surfaces with real multiplication

Seminar | September 11 | 3:10-5 p.m. | 891 Evans Hall

 Yunqing Tang, Princeton University

 Department of 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 $\mathfrak p$ 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.

Seminar Format: The seminar consists of two 50-minute talks, a pre-talk (3:10-4:00) and an advanced talk (4:10-5:00), with a 10-minute break (4:00-4:10) between them. The advanced talk is a regular formal presentation about recent research results to general audiences in arithmetic geometry and number theory; the pre-talk (3:10-4:00) is to introduce some prerequisites or background for the advanced talk to audiences consisting of graduate students.