\((\varphi,\Gamma)\)-modules and representations of the mirabolic group of \(\text{GL}_2(\mathbb Q_p))\).
(\((\varphi,\Gamma)\)-modules et représentations du mirabolique de \(\text{GL}_2(\mathbb Q_p))\).)

*(French)*Zbl 1235.11107
Berger, Laurent (ed.) et al., Représentations \(p\)-adiques de groupes \(p\)-adiques II: Représentations de \(\text{GL}_2 (\mathbb Q_p)\) et \((\varphi, \gamma)\)-modules. Paris: Société Mathématique de France (ISBN 978-2-85629-281-5/pbk). Astérisque 330, 61-153 (2010).

This paper, along with its companion paper [the author, “Representations of \(\mathrm{GL}_2({\mathbb{Q}}_p)\) and \((\phi,\Gamma)\)-modules,” Astérisque 330, 281–509 (2010; Zbl 1218.11107)], are the crowning achievement of several years’ work by the author and others, which has led to a satisfactory extension of the local Langlands correspondence to cover all 2-dimensional \(p\)-adic representations of the Galois group of \(\mathbb{Q}_p\).

The focus of this article is on the construction of a functor – indeed two closely related functors – from \(p\)-adic Galois representations to representations of the “mirabolic subgroup” \(P(\mathbb{Q}_p) := \begin{pmatrix} \mathbb{Q}_p^\times & \mathbb{Q}_p \\ 0 & 1 \end{pmatrix}\) of \(\mathrm{GL}_2(\mathbb{Q}_p)\). This construction passes via Fontaine’s theory of étale \((\varphi, \Gamma)\)-modules, which associates to any \(p\)-adic Galois representation a vector space \(D\) over a certain field \(\mathcal{E}\) of formal Laurent series in a variable \(T\), with semilinear actions of a Frobenius \(\varphi\) and the group \(\Gamma = \mathbb{Z}_p^\times\). These data can be organized into an action of the semigroup \(P^+ = \begin{pmatrix} \mathbb{Z}_p - \{0\} & \mathbb{Z}_p \\ 0 & 1 \end{pmatrix}\) of \(P(\mathbb{Q}_p)\).

The guiding principle of the construction is that the trivial \((\varphi, \Gamma)\)-module \(\mathcal{E}\) contains a natural submodule \(\mathcal{E}^+\) isomorphic to the space of measures on \(\mathbb{Z}_p\), and the quotient \(\mathcal{E} / \mathcal{E}^+\) is isomorphic to the \(p\)-adic Banach space of continuous functions on \(\mathbb{Z}_p\), the dual of the space of measures. The author shows that a general étale \((\varphi, \Gamma)\)-module can be decomposed (in two related ways, using submodules \(D^\sharp, D^\natural\) differing by finite-dimensional pieces) as an extension of a Banach space by the dual of a Banach space. Moreover, by translating into \((\varphi, \Gamma)\)-module terms the definition of the space of measures on the whole of \(\mathbb{Q}_p\), the author defines modules \(D \boxtimes \mathbb{Q}_p\) and \(D^\natural \boxtimes \mathbb{Q}_p\) having an action of the whole mirabolic subgroup \(P(\mathbb{Q}_p)\); and these spaces have operations of “restriction to open subsets” (denoted \(D \boxtimes U\) for \(U \subseteq \mathbb{Q}_p\) open) and ”pushforward by a local diffeomorphism” modelled on the corresponding operations for functions and for measures. These constructions, and a related functor \(D \mapsto D \boxtimes \mathbb{P}^1(\mathbb{Q}_p)\), play a key role in the definition of the local Langlands correspondence in the following paper.

The final result of the paper is of a slightly different type. This gives a formula relating certain pairings on the submodules \(D \boxtimes \mathbb{Z}_p^\times\). The definition of these pairings is technical – see Theorem 0.8 of the paper – but when \(D\) is attached to a Galois representation which is crystalline in the sense of Fontaine, this statement recovers Perrin-Riou’s explicit reciprocity law (see [B. Perrin-Riou, “Théorie d’Iwasawa des représentations \(p\)-adiques sur un corps local,” Invent. Math. 115, No. 1, 81–149 (1994; Zbl 0838.11071)] and also [B. Perrin-Riou, “Théorie d’Iwasawa et loi explicite de réciprocité. Un remake d’un article de P. Colmez,” Doc. Math., J. DMV 4, 219–273 (1999; Zbl 0928.11046)]). This result is used in the next paper in order to establish the compatibility of the \(p\)-adic Langlands correspondence with the classical one.

This paper and the one following it are a monumental achievement in \(p\)-adic Hodge theory, and should be read by any mathematician seriously interested in the subject. It is very clearly written, with a thorough introduction and a detailed summary of the existing theory. Readers should, however, be warned that (despite its length) this first paper is not intended to be read in isolation: while it is technically self-contained, the significance of the results only becomes clear in the light of the applications made in the following even longer article. (A detailed description of the results of the present paper is given in §I of its sequel.)

For the entire collection see [Zbl 1192.11001].

The focus of this article is on the construction of a functor – indeed two closely related functors – from \(p\)-adic Galois representations to representations of the “mirabolic subgroup” \(P(\mathbb{Q}_p) := \begin{pmatrix} \mathbb{Q}_p^\times & \mathbb{Q}_p \\ 0 & 1 \end{pmatrix}\) of \(\mathrm{GL}_2(\mathbb{Q}_p)\). This construction passes via Fontaine’s theory of étale \((\varphi, \Gamma)\)-modules, which associates to any \(p\)-adic Galois representation a vector space \(D\) over a certain field \(\mathcal{E}\) of formal Laurent series in a variable \(T\), with semilinear actions of a Frobenius \(\varphi\) and the group \(\Gamma = \mathbb{Z}_p^\times\). These data can be organized into an action of the semigroup \(P^+ = \begin{pmatrix} \mathbb{Z}_p - \{0\} & \mathbb{Z}_p \\ 0 & 1 \end{pmatrix}\) of \(P(\mathbb{Q}_p)\).

The guiding principle of the construction is that the trivial \((\varphi, \Gamma)\)-module \(\mathcal{E}\) contains a natural submodule \(\mathcal{E}^+\) isomorphic to the space of measures on \(\mathbb{Z}_p\), and the quotient \(\mathcal{E} / \mathcal{E}^+\) is isomorphic to the \(p\)-adic Banach space of continuous functions on \(\mathbb{Z}_p\), the dual of the space of measures. The author shows that a general étale \((\varphi, \Gamma)\)-module can be decomposed (in two related ways, using submodules \(D^\sharp, D^\natural\) differing by finite-dimensional pieces) as an extension of a Banach space by the dual of a Banach space. Moreover, by translating into \((\varphi, \Gamma)\)-module terms the definition of the space of measures on the whole of \(\mathbb{Q}_p\), the author defines modules \(D \boxtimes \mathbb{Q}_p\) and \(D^\natural \boxtimes \mathbb{Q}_p\) having an action of the whole mirabolic subgroup \(P(\mathbb{Q}_p)\); and these spaces have operations of “restriction to open subsets” (denoted \(D \boxtimes U\) for \(U \subseteq \mathbb{Q}_p\) open) and ”pushforward by a local diffeomorphism” modelled on the corresponding operations for functions and for measures. These constructions, and a related functor \(D \mapsto D \boxtimes \mathbb{P}^1(\mathbb{Q}_p)\), play a key role in the definition of the local Langlands correspondence in the following paper.

The final result of the paper is of a slightly different type. This gives a formula relating certain pairings on the submodules \(D \boxtimes \mathbb{Z}_p^\times\). The definition of these pairings is technical – see Theorem 0.8 of the paper – but when \(D\) is attached to a Galois representation which is crystalline in the sense of Fontaine, this statement recovers Perrin-Riou’s explicit reciprocity law (see [B. Perrin-Riou, “Théorie d’Iwasawa des représentations \(p\)-adiques sur un corps local,” Invent. Math. 115, No. 1, 81–149 (1994; Zbl 0838.11071)] and also [B. Perrin-Riou, “Théorie d’Iwasawa et loi explicite de réciprocité. Un remake d’un article de P. Colmez,” Doc. Math., J. DMV 4, 219–273 (1999; Zbl 0928.11046)]). This result is used in the next paper in order to establish the compatibility of the \(p\)-adic Langlands correspondence with the classical one.

This paper and the one following it are a monumental achievement in \(p\)-adic Hodge theory, and should be read by any mathematician seriously interested in the subject. It is very clearly written, with a thorough introduction and a detailed summary of the existing theory. Readers should, however, be warned that (despite its length) this first paper is not intended to be read in isolation: while it is technically self-contained, the significance of the results only becomes clear in the light of the applications made in the following even longer article. (A detailed description of the results of the present paper is given in §I of its sequel.)

For the entire collection see [Zbl 1192.11001].

Reviewer: David Loeffler (Coventry)