Local $\epsilon$-isomorphisms for rank two $p$-adic representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$ 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 $\epsilon$-conjecture using the Colmez’s $p$-adic local Langlands correspondence for $\mathrm{GL}_2 (\mathbb{Q}_p)$. 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 $\mathrm{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$ 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 $\epsilon$-conjecture.

Keywords

$p$-adic Hodge theory, $(\varphi, \Gamma)$-module

2010 Mathematics Subject Classification

Primary 11F80. Secondary 11F85, 11S25.

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Received 14 January 2016

Published 7 August 2017