Stochastic Canonical Heights

TL;DR

This paper introduces stochastic height functions on projective varieties with endomorphisms, exploring their properties, kernels, and applications to dynamical systems and finiteness results over global fields.

Contribution

It constructs and analyzes stochastic canonical height functions, linking their kernels to Julia sets and applying them to finiteness problems in number theory.

Findings

Stochastic heights satisfy properties similar to classical canonical heights.

The kernel of stochastic heights relates to Julia sets in the projective line case.

Finiteness of certain generalized Zsigmondy sets over global fields is established.

Abstract

We construct height functions defined stochastically on projective varieties equipped with endomorphisms, and we prove that these functions satisfy analogs of the usual properties of canonical heights. Moreover, we give a dynamical interpretation of the kernel of these stochastic height functions, and in the case of the projective line, we relate the size of this kernel to the Julia sets of the original maps. Finally, as an application, we establish the finiteness of some generalized Zsigmondy sets over global fields.