Theta-induced diffusion on Tate elliptic curves over nonarchimedean local fields

Patrick Erik Bradley

Vol. 334 (2025), No. 1, 13–42

DOI: 10.2140/pjm.2025.334.13

Abstract

A diffusion operator on the $K$-rational points of a Tate elliptic curve $E_q$ is constructed, where $K$ is a nonarchimedean local field, as well as an operator on the Berkovich analytification $E_q^{\mathrm{an}<!--FUNCTION\; APPLICATION-->}$ of $E_q$. These are integral operators for measures coming from a regular $1$-form, and kernel functions constructed via theta functions. The second operator can be described via certain nonarchimedean curvature forms on $E_q^{\mathrm{an}<!--FUNCTION\; APPLICATION-->}$. The spectra of these self-adjoint bounded operators on the Hilbert spaces of $L^2$-functions are identical and found to consist of finitely many eigenvalues. A study of the corresponding heat equations yields a positive answer to the Cauchy problem, and induced Markov processes on the curve. Finally, some geometric information about the $K$-rational points of $E_q$ is retrieved from the spectrum.

Keywords

Tate curve, $p$-adic diffusion, theta function, Laplacian, torsion points

Mathematical Subject Classification

Primary: 14H52

Secondary: 58J35

Milestones

Received: 19 February 2024

Revised: 15 October 2024

Accepted: 24 December 2024

Published: 30 December 2024