Product Formulas for Periods of CM Abelian Varieties and the Function Field Analog

TL;DR

This paper surveys Colmez's conjecture on Faltings heights and period formulas for CM abelian varieties, and explores the analogous theory in the function field setting involving Drinfeld modules and A-motives.

Contribution

It provides a comprehensive overview of the classical theory and develops the function field analog, connecting abelian varieties with Drinfeld modules and A-motives.

Findings

Survey of Colmez's conjecture and theory on periods and heights.

Development of the function field analog involving Drinfeld modules.

Explanation of cohomology theories and comparison isomorphisms in both settings.

Abstract

We survey Colmez's theory and conjecture about the Faltings height and a product formula for the periods of abelian varieties with complex multiplication, along with the function field analog developed by the authors. In this analog, abelian varieties are replaced by Drinfeld modules and $A$-motives. We also explain the necessary background on abelian varieties, Drinfeld modules and $A$-motives, including their cohomology theories and comparison isomorphisms and their theory of complex multiplication.