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ON THE GALOIS STRUCTURE OF ARITHMETIC COHOMOLOGY I: COMPACTLY SUPPORTED $p$-ADIC COHOMOLOGY

Part of: Algebraic number theory: global fields Representation theory of groups

Published online by Cambridge University Press:  16 November 2018

DAVID BURNS
DAVID BURNS*
Affiliation:
King’s College London, Department of Mathematics, London WC2R 2LS, UK email david.burns@kcl.ac.uk
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Abstract

We investigate the Galois structures of $p$-adic cohomology groups of general $p$-adic representations over finite extensions of number fields. We show, in particular, that as the field extensions vary over natural families the Galois modules formed by these cohomology groups always decompose as the direct sum of a projective module and a complementary module of bounded $p$-rank. We use this result to derive new (upper and lower) bounds on the changes in ranks of Selmer groups over extensions of number fields and descriptions of the explicit Galois structures of natural arithmetic modules.

MSC classification

Primary: 11R34: Galois cohomology 11R23: Iwasawa theory 20C11: $p$-adic representations of finite groups

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Type
Article
Information
Nagoya Mathematical Journal , Volume 239 , September 2020 , pp. 294 - 321
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Copyright
© 2018 Foundation Nagoya Mathematical Journal

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