Multivariable \((\varphi ,\Gamma)\)-modules and locally analytic vectors (Q504019)

The goal of this paper is to give one generalization of the classical \((\varphi, \Gamma)\)-module theory to \((\varphi_q, \Gamma)\)-theory for a Lubin-Tate tower. The ultimate hope behind this generalization is to shed some light on the \(p\)-adic local Langlands correspondence beyond the case of \(\mathrm{GL}_2(\mathbb Q_p)\). Let \(p\) be a prime number. The classical \((\varphi, \Gamma)\)-module theory harvested from the structure of the \(p\)-adic cyclotomic extension of a \(p\)-adic local field \(K\), to establish a canonical correspondence between the \(p\)-adic representations of \(K\) and the étale \((\varphi, \Gamma)\)-modules over \(\mathbf B_K\). This plays the key role in Colmez's proof of \(p\)-adic local Langlands correspondence for \(\mathrm{GL}_2(\mathbb Q_p)\) in \textit{P. Colmez} [in: Représentations \(p\)-adiques de groupes p-adiques II: Représentations de \(\mathrm{GL}_2(\mathbb Q_p)\) et \((\varphi, \Gamma)\)-modules. Paris: Société Mathématique de France. 281--509 (2010; Zbl 1218.11107)]. To go beyond the case of \(\mathrm{GL}_2(\mathbb Q_p)\), say the case of \(\mathrm{GL}_2(F)\) for a finite extension \(F\) of \(\mathbb Q_p\), the author suggests that one might try to consider the Lubin-Tate extension \(F_\infty\) of \(F\) defined by a uniformizer \(\pi\) of \(F\). For a finite extension \(K\) of \(F\), write \(K_\infty = KF_\infty\). Let \(V\) be a \(p\)-adic representation of the Galois group \(G_K\) of \(K\). The author suggests to look at the subspace \[ \widetilde{ \mathrm{D}}^\dagger_{\mathrm{rig}, K}(V)^{\mathrm{pa}}: = \big( \widetilde {\mathbf{B}}_{\mathrm{rig}}^\dagger \otimes V \big)^{H_K, \mathrm{pa}} \] consisting of pro-analytic vectors. The main result of this paper is that \(\widetilde{ \mathrm{D}}^\dagger_{\mathrm{rig}, K}(V)^{\mathrm{pa}}\) is a finite free module over \((\widetilde{ \mathbf{B}}^\dagger_{\mathrm{rig}, K})^{\mathrm{pa}}\) of rank \(\dim V\), and is stable under the action of \(\varphi\) and \(\Gamma_K: = \mathrm{Gal}(K_\infty/K)\). The author actually proved this for a more general class of \(p\)-adic Lie extension \(K_\infty\) of \(K\). The author also pointed out that when \(F=\mathbb Q_p\), one has \(\widetilde{ \mathrm{D}}^\dagger_{\mathrm{rig}, K}(V)^{\mathrm{pa}} =\cup_{n \geq 0} \varphi^{-n}(\mathrm{D}^\dagger_{\mathrm{rig}, K}(V))\), which is frequently used in Colmez's work. By this the author suggests that this might shed some lights on the structure of the highly speculative \(p\)-adic local Langlands for \(\mathrm{GL}_2(F)\). Using this main result, the author deduces three theorems on \(F\)-analytic representations. (1) He first proves a structure result on \(F\)-proanalytic vectors: \[ (\widetilde{ \mathbf{B}}^\dagger_{\mathrm{rig}, K})^{F\text{-}\mathrm{pa}} = \bigcup_{n \geq 0} \varphi_q^{-n}( \mathbf{B}^\dagger_{\mathrm{rig},K}), \] where \(\varphi_q\) is the \(q\)-Frobenius map induced by the multiplication by \(\pi\) of the formal group law. (2) The author also proves that the Lubin-Tate \((\varphi_q, \Gamma_K)\)-modules of \(F\)-analytic representations are overconvergent, where \(F\)-analytic means that \(\mathbb C_p \otimes_{F, \tau} V\) is the trivial semilinear representation with coefficients in \(\mathbb C_p\), for any \(\tau \neq \mathrm{id}\). This generalizes known results when \(F=\mathbb Q_p\) due to \textit{F. Cherbonnier} and \textit{P. Colmez} [Invent. Math. 133, No. 3, 581--611 (1998; Zbl 0928.11051)], and when \(V\) is \(F\)-crystalline due to \textit{M. Kisin} and \textit{W. Ren} [Doc. Math., J. DMV 14, 441--461 (2009; Zbl 1246.11112)]. (3) Finally, the author shows that the functor \(V \mapsto \mathrm{D}_{\mathrm{rig}}^\dagger(V)\) gives an equivalence of categories between the category of \(F\)-analytic representations of \(G_K\) and the category of étale \(F\)-analytic Lubin-Tate \((\varphi_q, \Gamma_K)\)-modules over \(\mathbf{B}_{\mathrm{rig}, K}^\dagger\).