Monday, October 31, 2022 — 2:30 PM EDT

Joseph H. Silverman, Brown University

"Finite Orbits of Points on Surfaces that Admit Three Non-commuting Involutions"

The classical Markoff-Hurwitz equation

M : x^2 + y^2 + z^2 = axyz + b

admits three non-commuting involutions coming from the three double covers M --> A^2. There has recently been considerable interest in studying the orbit structure of the (Z/pZ)-points of M under the action of the involutions. In this talk I will discuss some of this history, and then describe analogous results and conjectures on K3 surfaces W in P^1xP^1xP^1 given by the vanishing of a (2,2,2) form. Just as with the Markoff-Hurwitz surface, the three projections W --> P^1xP^1 are double covers that induce three non-commuting involutions on W. Let G be the group of automorphisms of W generated by these involutions. We investigate the G-orbit structure of the points of W. In particular, we study G-orbital components of W(Z/pZ) and finite G-orbits in W(C). This nice blend of number theory, geometry, and dynamics, requires no pre-requisites beyond an undergraduate algebra course. (This is joint work with Elena Fuchs, Matthew Litman, and Austin Tran.)

MC 5501

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