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Forum Mathematicum

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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


We introduce a certain Artin–Schreier scheme over a finite field associated to a pair of coprime integers (m,n) with n3 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.

Keywords: Artin–Schreier varieties; Heisenberg groups

MSC 2010: 14R20; 14F20


  • [1]

    D. Bump, Automorphic Forms and Representations, Cambridge Stud. Adv. Math. 55, Cambridge University Press, Cambridge, 1998. Google Scholar

  • [2]

    C. J. Bushnell and G. Henniart, The essentially tame local Langlands correspondence. II: Totally ramified representations, Compos. Math. 141 (2005), no. 4, 979–1011. CrossrefGoogle Scholar

  • [3]

    C. J. Bushnell and G. Henniart, The essentially tame Jacquet–Langlands correspondence for inner forms of GL(n), Pure Appl. Math. Q. 7 (2011), no. 3, 469–538. Google Scholar

  • [4]

    C. J. Bushnell and P. C. Kutzko, The Admissible Dual of GL(N) via Compact Open Subgroups, Ann. of Math. Stud. 129, Princeton University Press, Princeton, 1993. Google Scholar

  • [5]

    P. Deligne, Cohomologie étale. Séminaire de Géométrie Algébrique du Bois-Marie SGA 412, Lecture Notes in Math. 569, Springer, Berlin, 1977. Google Scholar

  • [6]

    P. Deligne, La conjecture de Weil II, Publ. Math. Inst. Hautes Études Sci. 52 (1980), 137–252. CrossrefGoogle Scholar

  • [7]

    J. Denef and F. Loeser, Character sums associated to finite Coxeter groups, Trans. Amer. Math. Soc. 350 (1998), no. 12, 5047–5066. CrossrefGoogle Scholar

  • [8]

    N. M. Katz, Sommes exponentielles, Astérisque 79 (1980), 1–209. Google Scholar

  • [9]

    N. M. Katz, Moments, Monodromy, and Perversity. A Diophantine Perspective, Ann. of Math. Stud. 159, Princeton University Press, Princeton, 2005. Google Scholar

  • [10]

    K. Tokimoto, Affinoids in the Lubin–Tate perfectoid space and special cases of the local Langlands correspondence in positive characteristic, preprint (2016), https://arxiv.org/abs/1609.02524.

About the article

Received: 2017-04-17

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, DOI: https://doi.org/10.1515/forum-2017-0085.

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