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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Frobenius actions on the de Rham cohomology of Drinfeld modules


Author: Ernst-Ulrich Gekeler
Journal: Trans. Amer. Math. Soc. 363 (2011), 3167-3183
MSC (2010): Primary 11G09; Secondary 11R58
Published electronically: January 27, 2011
MathSciNet review: 2775802
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Abstract: We study the action of endomorphisms of a Drinfeld $ A$-module $ \phi$ on its de Rham cohomology $ H_{DR}(\phi,L)$ and related modules, in the case where $ \phi$ is defined over a field $ L$ of finite $ A$-characteristic $ \frak p$. Among others, we find that the nilspace $ H_0$ of the total Frobenius $ Fr_{DR}$ on $ H_{DR}(\phi,L)$ has dimension $ h = $ height of $ \phi$. We define and study a pairing between the $ \frak p$-torsion $ _{\frak p}\phi$ of $ \phi$ and $ H_{DR}(\phi,L)$, which becomes perfect after dividing out $ H_0$.


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Additional Information

Ernst-Ulrich Gekeler
Affiliation: FR 6.1 Mathematik, E2 4, Universität des Saarlandes, Postfach 15 11 50, D-66041 Saarbrücken, Germany
Email: gekeler@math.uni-sb.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05422-X
PII: S 0002-9947(2011)05422-X
Keywords: Drinfeld module, de Rham cohomology
Received by editor(s): July 20, 2009
Published electronically: January 27, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.