Periods of Drinfeld modules and local shtukas with complex multiplication

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.

v2: 28 pages, final version, appeared in J. Inst. Math. Jussieu 19 (2020), no. 1, 175--208, v3, v4: 32 pages, final version plus an erratum in appendix B

References in corpus (5)

• Integral p-adic Hodge theory
• Local Shtukas and Divisible Local Anderson Modules
• Local Shtukas, Hodge-Pink Structures and Galois Representations
• Periods of Drinfeld modules and local shtukas with complex multiplication
• Pink's theory of Hodge structures and the Hodge conjecture over function fields

Cited by in corpus (2)

• Periods of Drinfeld modules and local shtukas with complex multiplication
• Product Formulas for Periods of CM Abelian Varieties and the Function Field Analog