Mathematics Colloquium

For an elliptic curve E over the rational numbers, the work of Lang and Trotter provides heuristics for the behavior of the reductions of E modulo primes. The Lang-Trotter philosophy applies to abelian varieties, which are higher dimensional generalizations of elliptic curves, and predicts that some density zero sets of primes with certain reduction types ought to be infinite.

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. Moreover, I will present an application of our results to the distribution of Hecke orbits in positive characteristic fields. These results are joint work with Ananth Shankar, with Ananth Shankar, Arul Shankar, and Salim Tayou, and with Davesh Maulik and Ananth Shankar.