Local epsilon isomorphisms

Abstract

In this paper, we prove the “local $\epsilon$-isomorphism” conjecture of Fukaya and Kato for a particular class of Galois modules, obtained by interpolating the twists of a fixed crystalline representation of $G_{{\mathbf{Q}}_p}$ by a family of characters of $G_{{\mathbf{Q}}_p}$. This can be regarded as a local analogue of the Iwasawa main conjecture for abelian $p$-adic Lie extensions of ${\mathbf{Q}}_p$, extending earlier work of Kato for rank one modules and of Benois and Berger for the cyclotomic extension. We show that such an $\epsilon$-isomorphism can be constructed using the 2-variable version of the Perrin-Riou regulator map constructed by the first and third authors.