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Local \(\varepsilon\)-isomorphisms for rank two \(p\)-adic representations of \(\operatorname{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)\) and a functional equation of Kato’s Euler system. (English) Zbl 1426.11055

Summary: In this article, we prove many parts of the rank two case of the Kato’s local \(\varepsilon\)-conjecture using the Colmez’s \(p\)-adic local Langlands correspondence for \(\operatorname{GL}_2 (\mathbb{Q}_p)\). We show that a Colmez’s pairing defined in his study of locally algebraic vectors gives us the conjectural \(\varepsilon\)-isomorphisms for (almost) all the families of \(p\)-adic representations of \(\operatorname{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)\) 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 \(\varepsilon\)-conjecture.

MSC:

11F80 Galois representations
11F85 \(p\)-adic theory, local fields
11S25 Galois cohomology
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