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PERIODS OF DRINFELD MODULES AND LOCAL SHTUKAS WITH COMPLEX MULTIPLICATION

Published online by Cambridge University Press:  20 March 2018

Urs Hartl
Affiliation:
Universität Münster, Mathematisches Institut, Einsteinstr. 62, D – 48149 Münster, GermanyURL: www.math.uni-muenster.de/u/urs.hartl/
Rajneesh Kumar Singh
Affiliation:
Ramakrishna Mission Vivekananda University, PO Belur Math, Dist Howrah 711202, West Bengal, India

Abstract

Colmez [Périodes des variétés abéliennes a multiplication complexe, Ann. of Math. (2)138(3) (1993), 625–683; available at http://www.math.jussieu.fr/∼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.

Type
Research Article
Copyright
© The Author(s) 2018. Published by Cambridge University Press 

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Footnotes

Both authors acknowledge support by the Deutsche Forschungsgemeinschaft (DFG) in form of SFB 878.

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