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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 GL_2(ℚ_p). We show that a Colmez’s pairing defin...ed 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/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 which is supercuspidal at p satisfies a functional equation which has the same form as predicted by the Kato’s global ε-conjecture.続きを見る
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