# NTS ABSTRACTSpring2017

## Feb 1

 Yunqing Tang Exceptional splitting of reductions of abelian surfaces with real multiplication Abstract: 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 \$\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.