Local epsilon-isomorphisms for rank two p-adic representations of Gal(overline{Q}_p/Q_p) and a functional equation of Kato's Euler system

TL;DR

This paper proves parts of Kato's local epsilon-conjecture for rank two p-adic Galois representations using Colmez's p-adic local Langlands correspondence, establishing epsilon-isomorphisms and a functional equation for Kato's Euler system.

Contribution

It demonstrates the construction of epsilon-isomorphisms for rank two p-adic Galois representations via Colmez's pairing, extending to critical Hodge-Tate weights and proving a functional equation for Kato's Euler system.

Findings

Established epsilon-isomorphisms for de Rham and trianguline cases.

Proved the functional equation for Kato's Euler system.

Extended interpolation properties to critical Hodge-Tate weights.

Abstract

In this article, we prove (many parts of) the rank two case of the Kato's local epsilon-conjecture using the Colmez's p-adic local Langlands correspondence for GL_2(Q_p). We show that a Colmez's pairing defined in his study of locally algebraic vectors gives us the conjectural epsilon-isomorphisms for (almost) all the families of p-adic representations of Gal(overline{Q}_p/Q_p) of rank two, which satisfy the desired interpolation property for the de Rham and trianguline case. For the de Rham and non trianguline case, we also show this interpolation property for the "critical" range of Hodge-Tate weights using the Emerton's theorem on the compatibility of classical and p-adic local Langlands correspondence. As an application, we prove that the Kato's Euler system associated to any Hecke eigen new form satisfies a functional equation which has the same form as predicted by the Kato's global epsilon-conjecture.