Periods of Drinfeld modules and local shtukas with complex multiplication

TL;DR

This paper proves a product formula for periods of CM $A$-motives in function fields, analogous to Colmez's conjecture for abelian varieties over number fields, by analyzing local shtukas and Artin $L$-series.

Contribution

It establishes the product formula for the Carlitz module and computes valuations of periods of CM $A$-motives at all finite places using local shtukas and Artin $L$-series.

Findings

Proved the product formula for the Carlitz module.

Computed valuations of periods at finite places.

Linked valuations to Artin $L$-series.

Abstract

Colmez conjectured a product formula for periods of abelian varieties over number fields with complex multiplication and proved it in some cases. His conjecture is equivalent to a formula for the Faltings height of CM abelian varieties in terms of the logarithmic derivatives at $s=0$ of certain Artin $L$-functions. In a series of articles we investigate the analog of Colmez's theory in the arithmetic of function fields. There abelian varieties are replaced by Drinfeld modules and their higher dimensional generalizations, so-called $A$-motives. In the present article we prove the product formula for the Carlitz module and we compute the valuations of the periods of a CM $A$-motive at all finite places in terms of Artin $L$-series. The latter is achieved by investigating the local shtukas associated with the $A$-motive.