Abstract

The arithmeticity of automorphism groups of algebraic varieties is a natural question at the intersection of number theory, algebraic geometry, and hyperbolic geometry. A basic problem is to determine when the automorphism group of a given variety is an arithmetic group.For K3 surfaces, Hashimoto--Lee constructed examples of Picard rank 3 whose automorphism groups are isomorphic to modular groups. On the other hand, Totaro exhibited K3 surfaces of Picard rank 4 whose automorphism groups are not arithmetic. Thus Picard rank 3/4 can be viewed as a borderline case for the arithmeticity of automorphism groups of K3 surfaces.In this talk, I will present joint work with Hashimoto in which we construct analogous examples of Picard rank  K3 surfaces whose automorphism groups are arithmetic. More precisely, these automorphism groups admit a natural realization as arithmetic subgroups of matrix groups defined over imaginary quadratic fields. I will explain the construction of these K3 surfaces and the resulting connection between their geometry and the associated arithmetic groups.

Zoom Link

https://us02web.zoom.us/j/83608621392?pwd=GhiB8iHs6ccDCYrS77Tq3EAMmfxwe9.1

Orgainizer

Weichung Chen (NTU)