Diffusion operators on $p$-adic analytic manifolds

TL;DR

This paper constructs and analyzes diffusion operators on p-adic analytic manifolds, establishing their spectral properties, associated heat kernels, and applications to counting points on elliptic curves over finite fields.

Contribution

It introduces Vladimirov-Taibleson type operators on p-adic manifolds, studies their spectra, and connects them to counting points on elliptic curves, providing new analytical tools in p-adic geometry.

Findings

Established the L^2-spectrum of the operator.

Proved the existence of a heat kernel and Green function for s > 1.

Applied the operator to count points on elliptic curves over finite fields.

Abstract

Kernel functions for Laplacian integral operators are constructed on $p$-adic analytic manifolds using charts and transition maps from an atlas with connected nerve complex. In the compact case, an operator of Vladimirov-Taibleson type parametrised by a real parameter $s$ is defined. Its kernel function uses a geodetic-like distance function on the nerve complex of its atlas. The $L^2$-spectrum of this operator is established, and it is shown that it gives rise to a Feller semigroup. In this way, the Cauchy problem for the corresponding heat equation is solved in the positive by a transition function of a Markov process. The existence of a heat kernel function and a Green function in the case $s > 1$ is proven. As an application, it is shown how to express the number of points on the reduction curve defined over the residue field of an elliptic curve with good reduction in terms of the eigenvalues of a Vladimirov-Taibleson-like operator. This provides for an alternative way of counting points on elliptic curves defined over finite fields.