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Wach modules, regulator maps and epsilon-isomorphisms in families

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Title: Wach modules, regulator maps and epsilon-isomorphisms in families
Authors: Bellovin, RM
Venjakob, O
Item Type: Journal Article
Abstract: We prove the “local ε-isomorphism” conjecture of Fukaya and Kato [13] for certain crystalline families of GQp-representations. This conjecture can be regarded as a local analog of the Iwasawa main conjecture for families. Our work extends earlier work of Kato for rank-1 modules (cf. [33]), of Benois and Berger for crystalline GQp-representations with respect to the cyclotomic extension (cf. [1]), as well as of Loeffler et al. (cf. [21]) for crystalline GQp-representations with respect to abelian p-adic Lie extensions of Qp⁠. Nakamura [24, 25] has also formulated a version of Kato’s ε-conjecture for affinoid families of (φ,Γ)-modules over the Robba ring, and proved his conjecture in the rank-1 case. He used this case to construct an ε-isomorphism for families of trianguline (φ,Γ)-modules, depending on a fixed triangulation. Our results imply that this ε-isomorphism is independent of the chosen triangulation for certain crystalline families. The main ingredient of our proof consists of the construction of families of Wach modules generalizing work of Wach and Berger [6] and following Kisin’s approach to the construction of potentially semi-stable deformation rings [18].
Issue Date: Aug-2019
Date of Acceptance: 19-Oct-2017
URI: http://hdl.handle.net/10044/1/53751
DOI: https://doi.org/10.1093/imrn/rnx276
ISSN: 1073-7928
Publisher: Oxford University Press (OUP)
Start Page: 5127
End Page: 5204
Journal / Book Title: International Mathematics Research Notices
Volume: 2019
Issue: 16
Copyright Statement: This is a pre-copyedited, author-produced PDF of an article accepted for publication in International Mathematics Research Notices following peer review. The version of record Rebecca Bellovin, Otmar Venjakob, Wach Modules, Regulator Maps, and ε-Isomorphisms in Families, International Mathematics Research Notices, Volume 2019, Issue 16, August 2019, Pages 5127–5204 is available online at: https://dx.doi.org/10.1093/imrn/rnx276
Keywords: 0101 Pure Mathematics
General Mathematics
Publication Status: Published
Online Publication Date: 2017-11-06
Appears in Collections:Faculty of Natural Sciences