Local epsilon isomorphisms

TL;DR

This paper proves a specific case of the local epsilon-isomorphism conjecture for certain Galois modules, extending previous work and utilizing Perrin-Riou regulator maps to establish the isomorphism.

Contribution

It establishes the local epsilon-isomorphism conjecture for a class of Galois modules related to crystalline representations and abelian quotients, extending prior results.

Findings

Construction of epsilon-isomorphisms using Perrin-Riou regulator maps.

Extension of the local epsilon-isomorphism conjecture to new classes of Galois modules.

Connection to the local analogue of the Iwasawa main conjecture.

Abstract

In this paper, we prove the "local epsilon-isomorphism conjecture" of Fukaya and Kato for a particular class of Galois modules obtained by tensoring a Zp-lattice in a crystalline representation of the Galois group of Qp with a representation of an abelian quotient of the Galois group with values in a suitable p-adic local ring. This can be regarded as a local analogue of the Iwasawa main conjecture for abelian p-adic Lie extensions of Qp, extending earlier work of Benois and Berger for the cyclotomic extension. We show that such an epsilon-isomorphism can be constructed using the Perrin-Riou regulator map, or its extension to the 2-variable case due to the first and third authors.