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COHOMOLOGY AND OVERCONVERGENCE FOR REPRESENTATIONS OF POWERS OF GALOIS GROUPS

Part of: Algebraic number theory: local and $p$-adic fields Discontinuous groups and automorphic forms Homological methods (field theory)

Published online by Cambridge University Press:  11 April 2019

Aprameyo Pal  and
Gergely Zábrádi Open the ORCID record for Gergely Zábrádi [Opens in a new window]
Aprameyo Pal
Affiliation:
Universität Duisburg-Essen, Fakultät für Mathematik, Thea-Leymann-Straße 9, D-45127Essen, Germany ([email protected])
Gergely Zábrádi
Affiliation:
Eötvös Loránd University, Institute of Mathematics, Pázmány Péter sétány 1/C, H-1117Budapest, Hungary Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, BudapestH-1364, Hungary MTA Rényi Intézet Lendület Automorphic Research Group, Hungary ([email protected])
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Abstract

We show that the Galois cohomology groups of $p$-adic representations of a direct power of $\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$ can be computed via the generalization of Herr’s complex to multivariable $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules. Using Tate duality and a pairing for multivariable $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules we extend this to analogues of the Iwasawa cohomology. We show that all $p$-adic representations of a direct power of $\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$ are overconvergent and, moreover, passing to overconvergent multivariable $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules is an equivalence of categories. Finally, we prove that the overconvergent Herr complex also computes the Galois cohomology groups.

Keywords

(𝜑, 𝛤)-modulesp-adic Galois representationsoverconvergenceHerr’s complex

MSC classification

Primary: 11S25: Galois cohomology
Secondary: 11S20: Galois theory 11S37: Langlands-Weil conjectures, nonabelian class field theory 11F80: Galois representations 12G05: Galois cohomology
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 2 , March 2021 , pp. 361 - 421
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Copyright
© Cambridge University Press 2019

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