Towards the Colmez Conjecture

TL;DR

This paper advances the understanding of Colmez's conjecture by establishing new links between periods of CM abelian varieties, heights of special points, and L-functions, proposing a geometric reformulation and explicit conjectures.

Contribution

It introduces a new definition for the height of a partial CM-type, proves the conjecture from an arithmetic period formula, and relates periods to Hermitian line bundles.

Findings

Colmez conjecture follows from an arithmetic period formula for surfaces

Explicit conjecture relating heights of special points to L-values

Heights of periods linked to arithmetic degrees of Hermitian line bundles

Abstract

We prove a collection of results involving Colmez's periods and the Colmez Conjecture. Using Colmez's theory of periods of CM abelian varieties, we propose a definition for the height of a partial CM-type and prove that the Colmez conjecture follows from an arithmetic period formula for surfaces. We give an explicit conjecture for the form of this period formula, which relates the height of special points on a Shimura surface with special values of $L$-functions. Further, we relate the heights of periods given by Colmez to arithmetic degree of Hermitian line bundles and thus give a formulation of Colmez's full conjecture in geometric terms.