Local heights on Galois covers of the projective line

TL;DR

This paper introduces a new local canonical height on algebraic curves over number fields, generalizing Tate's function, and relates it to potential theory on Berkovich curves, with applications to Galois covers.

Contribution

It defines a local height on Galois covers of the projective line using potential theory, extending previous work and connecting it to Mahler measures.

Findings

The local height is expressed as an integral with a logarithmic integrand.

The height can be obtained by averaging over higher order Weierstrass points.

The construction generalizes previous results by Everest-ni Fhlathuin and Szpiro-Tucker.

Abstract

Let X be a smooth projective curve of positive genus defined over a number field K. Assume given a Galois covering map x from X to the projective line over K and a place v of K. We introduce a local canonical height on the set of K_v-valued points of X associated to x as an integral with logarithmic integrand, generalizing Tate's local Neron function on an elliptic curve. The resulting global height can be viewed as a 'Mahler measure' associated to x. We prove that the local canonical height can be obtained by averaging, and taking a limit, over divisors of higher order Weierstrass points on X. This generalizes previous results by Everest-ni Fhlathuin and Szpiro-Tucker. Our construction of the local canonical height is an application of potential theory on Berkovich curves in the presence of a canonical measure.