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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 31, Issue 1

# On middle cohomology of special Artin–Schreier varieties and finite Heisenberg groups

Takahiro Tsushima
• Corresponding author
• Department of Mathematics and Informatics, Faculty of Science, Chiba University 1-33 Yayoi-cho, Inage, Chiba, 263-8522, Japan
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Published Online: 2018-09-10 | DOI: https://doi.org/10.1515/forum-2017-0085

## Abstract

We introduce a certain Artin–Schreier scheme over a finite field associated to a pair of coprime integers $\left(m,n\right)$ with $n\ge 3$ divisible by the characteristic of the base field, and study the middle étale cohomology group of it. If m is even, the variety admits actions of some finite Heisenberg groups. We study the middle cohomology as representations of the Heisenberg groups. If m is odd, we compute the Frobenius eigenvalues of it concretely. This affine scheme comes from the reduction of a certain affinoid in a Lubin–Tate space.

MSC 2010: 14R20; 14F20

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Revised: 2018-06-18

Published Online: 2018-09-10

Published in Print: 2019-01-01

Funding Source: Japan Society for the Promotion of Science

Award identifier / Grant number: 15K17506

This work was supported by JSPS KAKENHI Grant Number 15K17506.

Citation Information: Forum Mathematicum, Volume 31, Issue 1, Pages 83–110, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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