Towards the Colmez Conjecture
Roy Zhao
TL;DR
The article develops a framework to study Colmez's CM-period height conjecture by introducing a height $ht$ on CM-type class functions and a height $h( extphi)$ for partial CM-types. It proves that Colmez’s conjecture reduces to an explicit arithmetic period formula on Shimura surfaces and establishes a geometric reformulation via Arakelov degrees of line bundles, connecting period-based heights with Faltings-type heights. A central contribution is a detailed comparison between Colmez's height $ht( extPhi, au)$ and Yuan–Zhang's geometric height $h( extPhi, au)$, showing they differ by a computable constant depending only on the CM-field $E$, namely $h( extPhi, au)=ht( extPhi, au)+ rac{1}{2}\log(2 ext{ extpi})+ rac{1}{4g}\, ext{log}|d_E|- extmu_{Art}(a_{ extPhi, au}^0)$. This comparison, combined with a reduction to 2-element partial CM-types, yields a path to proving Colmez’s conjecture in full by verifying corresponding surface-period identities, and it confirms that nearby CM-types have heights governed solely by the CM-field, aligning the period and Arakelov viewpoints. The results further intertwine arithmetic of CM fields, L-function derivatives, and Shimura-geometry in a way that supports broader applications in André–Oort-type problems.
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.