A p-adic Laplacian on the Tate curve

11/27/2023 10:30 am - 11:30 am
CMSA Room G10
Address: CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Algebraic Geometry in String Theory Seminar

Speaker: An Huang (Brandeis)

Pre-talk Speaker: TBA: 10:00-10:30 am

Title: A p-adic Laplacian on the Tate curve

Abstract: We shall first explain the relation between a family of deformations of genus zero p-adic string worldsheet action and Tate’s thesis. We then propose a genus one p-adic string worldsheet action. The key is the definition of a p-adic Laplacian operator on the Tate curve. We show that the genus one p-adic Green’s function exists, is unique under some obvious constraints, is locally constant off diagonal, and has a reflection symmetry. It can also be numerically computed exactly off the diagonal, thanks to some simplifications due to the p-adic setup. Numerics suggest that at least in some special cases, the asymptotic behavior of the Green’s function near the diagonal is a direct p-adic counterpart of the familiar Archimedean case, although the p-adic Laplacian is not a local operator. Joint work in progress with Rebecca Rohrlich.