Theta-induced diffusion on Tate elliptic curvesover nonarchimedean local fields

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 an q 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 an q .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.