Canonical heights for correspondences

TL;DR

This paper extends the concept of canonical heights from polarized endomorphisms to polarized correspondences on projective varieties, establishing fundamental properties and a decomposition into local heights.

Contribution

It generalizes the canonical height construction to correspondences and demonstrates its fundamental properties and local-global decomposition.

Findings

Generalized canonical height for correspondences

Proved fundamental properties of the generalized height

Established local-global height decomposition

Abstract

The canonical height associated to a polarized endomporhism of a projective variety, constructed by Call and Silverman and generalizing the N\'eron-Tate height on a polarized Abelian variety, plays an important role in the arithmetic theory of dynamical systems. We generalize this construction to polarized correspondences, prove various fundamental properties, and show how the global canonical height decomposes as an integral of a local height over the space of absolute values on the algebraic closure of the field of definition.