Wach modules, regulator maps, and ε-isomorphisms in families

Bellovin, R. and Venjakob, O. (2019) Wach modules, regulator maps, and ε-isomorphisms in families. International Mathematics Research Notices, 2019(16), pp. 5127-5204. (doi: 10.1093/imrn/rnx276)

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, )-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, )- 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].

• Item Type: Articles
• Additional Information: This work was partially supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship award number [1401640] to R.B.; DFG-Forschergruppe award number [1920] "Symmetrie, Geometrie und Arithmetik" to O.V.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Bellovin, Dr Rebecca
• Authors: Bellovin, R., and Venjakob, O.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: International Mathematics Research Notices
• Publisher: Oxford University Press
• ISSN: 1073-7928
• ISSN (Online): 1687-0247