The conjecture of Colmez and reciprocity laws for modular forms

TL;DR

This paper refines Colmez's conjecture relating Faltings heights of CM abelian varieties to Artin L-functions and explores its implications for reciprocity laws in modular forms, proving it for elliptic curves.

Contribution

It proposes a refined conjecture linking transcendental quantities to Faltings heights and demonstrates its validity for elliptic curves, suggesting new reciprocity laws for modular forms.

Findings

Refined Colmez's conjecture for CM abelian varieties.

Proved the conjecture for elliptic curves.

Connected the conjecture to reciprocity laws for Siegel modular forms.

Abstract

In an article published in 1993, P. Colmez formulated a remarkable conjecture, which asserts that the Faltings height of a CM abelian variety can be computed as a linear combination of logarithmic derivatives of Artin $L$-functions. Noting that the Faltings height is an average of transcendental quantities summed over the embeddings of a number field of definition of the abelian variety, we propose a refinement of this conjecture, which identifies each of these transcendental quantities. We also show how our conjecture would imply the existence of fine reciprocity laws for Siegel modular forms with rational coefficients evaluated at CM points, and we prove our conjecture for elliptic curves, using old results of Siegel and Hasse on elliptic units.