Infinitesimal actions in the \(p\)-adic local Langlands correspondence
Let \(V\) be a two-dimensional absolutely irreducible \(\overline{\mathbb Q}_p\)-representation of the absolute Galois group \(G_{\mathbb Q_p}= \text{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)\). Let \(\Pi(V)\) be the \(\text{GL}_2(\mathbb Q_p)\) Banach representation associated by \textit{P. Colmez}'s \(p\)-adic Langlands correspondence [Astérisque 330, 281--509 (2010; Zbl 1218.11107)]. The author establishes a link between the action of the Lie algebra of \(\text{GL}_2(\mathbb Q_p)\) on the locally analytic vectors \(\Pi(V)^{\text{an}}\) of \(\Pi(V)\), the connection \(\nabla\) on the \((\phi,\Gamma)\)-module associated to \(V\) and the Sen polynomial of \(V\) (Theorem 1). This answers a question of Harris (see Remark 3.3.8 in [\textit{M. Emerton}, Pure Appl. Math. Q. 2, No. 2, 279--393 (2006; Zbl 1254.11106)]). Using this result the author gives a new (and more direct) proof of a theorem of Colmez (loc. cit., thms. VI 6.13 and VI 6.18), stating that \(\Pi(V)\) has non-zero locally algebraic vectors if and only if \(V\) is potentially semi-stable with distinct Hodge-Tate weights (Theorem 4). Let us stress that Theorem 4 is crucial in the proofs of (a part of) Fontaine-Mazur conjecture by \textit{M. Emerton} [Local-global compatibility in the \(p\)-adic Langlands programme for \(\text{GL}_2/\mathbb Q\), preprint] and by \textit{M. Kisin} [J. Am. Math. Soc. 22, No. 3, 641--690 (2009; Zbl 1251.11045)].
scientific article; zbMATH DE number 6087548
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