Frobenius actions on the de Rham cohomology of Drinfeld modules

Gekeler, Ernst-Ulrich

doi:10.1090/S0002-9947-2011-05422-X

Frobenius actions on the de Rham cohomology of Drinfeld modules
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by Ernst-Ulrich Gekeler

Trans. Amer. Math. Soc. 363 (2011), 3167-3183

DOI: https://doi.org/10.1090/S0002-9947-2011-05422-X

Published electronically: January 27, 2011

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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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