Algebraic Dynamics seminar: p-adic Siegel disks on K3 surfaces

McMullen constructed automorphisms of complex K3 surfaces with positive entropy—and thus chaotic dynamics on a large set—that nonetheless have a Siegel disk: an open region on which the automorphism is holomorphically conjugate to an irrational rotation. Strikingly, such an automorphism forces the K3 surface to be non-projective, so McMullen’s examples cannot be written down using explicit equations. Instead they are obtained indirectly via the global Torelli theorem.  In this talk we construct a p-adic analogue: a positive cohomological-entropy automorphism with a Siegel disk on a non-projective rigid-analytic K3 surface over a p-adic field. Our construction produces the surface directly using mod p geometry, without access to a Torelli theorem, which gives considerably more flexibility than in the complex case.