Local ε-isomorphisms for rank two -adic representations of Gal (ℚ^^―ₚ/ℚₚ) and a functional equation of kato's Euler system

In this article, we prove many parts of the rank two case of the Kato’s local ε-conjecture using the Colmez’s p-adic local Langlands correspondence for GL₂ (ℚₚ). We show that a Colmez’s pairing defined in his study of locally algebraic vectors gives us the conjectural ε-isomorphisms for (almost) all the families of p-adic representations of Gal (ℚ^^―ₚ/ℚₚ) 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 which is supercuspidal at p satisfies a functional equation which has the same form as predicted by the Kato’s global ε-conjecture.