Wednesday, January 15, 2020 - 4:15pm
Abstract: For an elliptic curve E over the rational numbers, the Sato—Tate conjecture and the Lang—Trotter philosophy provide heuristics for the behavior of the Frobenius endomorphisms of the reductions of E modulo primes. These heuristics predict that certain sets of primes of density zero ought to be infinite for every E. Similar heuristics apply to abelian varieties, which are higher dimensional generalizations of elliptic curves.
In this talk, I will discuss recent results that establish the infinitude of such sets for certain Kuga—Satake abelian varieties and K3 surfaces via intersection theory on their moduli spaces. In particular, for a K3 surface X over a number field with potentially good reduction everywhere, we prove that there are infinitely many primes modulo which the reduction of X has larger geometric Picard rank than that of the generic fiber X. These results are joint work with Ananth Shankar, with Ananth Shankar, Arul Shankar, and Salim Tayou, and with Davesh Maulik and Ananth Shankar.