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

TL;DR

This paper proves the local epsilon-isomorphism conjecture for certain crystalline G_Qp-representations, extending previous work and showing independence of triangulation choices in families, using Wach modules and regulator maps.

Contribution

It establishes the epsilon-isomorphism conjecture for crystalline families, generalizing prior results and constructing Wach modules in a new family context.

Findings

Proved the epsilon-isomorphism conjecture for specific crystalline families.

Constructed families of Wach modules generalizing previous work.

Showed independence of epsilon-isomorphism from triangulation choices.

Abstract

We prove the local epsilon-isomorphism conjecture of Fukaya and Kato [FK06] for certain crystalline families of G_Qp-representations. This conjecture can be regarded as a local analogue of the Iwasawa main conjecture for families. Our work extends earlier work of Kato for rank-1 modules (cf. [Ven13]), of Benois and Berger for crystalline G_Qp-representations with respect to the cyclotomic extension (cf. [BB08]), as well as of Loeffler, Venjakob, and Zerbes (cf. [LVZ13]) for crystalline G_Qp- representations with respect to abelian p-adic Lie extensions of Qp. Nakamura [Nak13, Nak14] has also formulated a version of the epsilon-conjecture for affinoid families of (phi,Gamma)-modules over the Robba ring, and proved his conjecture in the rank-1 case. He used this case to construct an epsilon-isomorphism for families of trianguline (phi,Gamma)-modules, depending on a fixed triangulation. Our results imply that this epsilon-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 [Ber04] and following the approach of Kisin to the construction of potentially semi-stable deformation rings [Kisa].