Remarks on some locally \(\mathbb Q_p\)-analytic representations of \(\mathrm{GL}_2(F)\) in the crystalline case (Q2900378)

This paper explains some expectations of the author on the \(p\)-adic local Langlands correspondence for \(\mathrm{GL}_2(F)\), where \(F\) is a finite extension of \(\mathbb Q_p\). The goal of the \(p\)-adic local Langlands correspondence is to attach certain \(p\)-adic representation of \(\mathrm{GL}_n(F)\) to an \(n\)-dimensional \(p\)-adic representation of \(\mathrm{Gal}(\overline{F}/F)\). This was known when \(n=2\) and \(F= \mathbb Q_p\) by the work of \textit{P. Colmez} [Astérisque 330, 281--509 (2010; Zbl 1218.11107)]. When \(n=2\) and \(F\neq \mathbb Q_p\), very little was known; this paper focuses on the \(\mathbb Q_p\)-locally analytic representations of \(\mathrm{GL}_2(F)\) associated to a crystalline representation \(V\) of \(\mathrm{Gal}(\overline{F}/ F)\) with distinct Hodge-Tate weights for each \(p\)-adic embedding of \(F\). Instead of making a guess on the entire representation (which might be untouchable at the current technique), the author constructs a locally analytic representation \(\Pi(V)\), which he conjectures to be a subrepresentation of the locally analytic representation that corresponds to \(V\).NEWLINENEWLINENEWLINETo a crystalline representation \(V\), the standard \(p\)-adic Hodge theory associates an admissible filtered \(\varphi\)-module \(D\). In fact, for each filtered \(\varphi\)-module \(D\), admissible or not, the author constructs a locally analytic representation \(\Pi(D)\) of \(\mathrm{GL}_2(F)\). For a ``generic'' \(D\), \(\Pi(D)\) is just the amalgamated sum of two natural locally \(\mathbb Q_p\)-analytic parabolic inductions associated to \(D\), over the locally algebraic vectors. But when the Hodge filtration on \(D\) is in a ``special position'' with respect to the eigenvectors of the Frobenius, the representation \(\Pi(D)\) has the same Jordan-Hölder factors as the amalgamated sum above, but some extensions (depending on positions of the Hodge filtration) inside the two parabolic induction are replaced by direct sums. For example, as a result, the socle of \(\Pi(D)\) may be larger than the locally algebraic representation. The author supports his expectations by proving the following three results.NEWLINENEWLINENEWLINE(1) If the socle of \(\Pi(D)\) admits a \(p\)-adic \(\mathrm{GL}_2(F)\)-invariant norm, then \(D\) is admissible, i.e. \(D\) comes from a crystalline representation \(V\) (this would be false in general if one simply uses the locally algebraic subrepresentation).NEWLINENEWLINENEWLINE(2) If the socle of \(\Pi(D)\) admits a \(p\)-adic \(\mathrm{GL}_2(F)\)-invariant norm, then its completion already contains a subrepresentation of \(\Pi(D)\) which is larger than the socle; this is a standard argument using Amice-Vélu and Vishik.NEWLINENEWLINENEWLINE(3) When \(D\) comes from a reducible crystalline representation \(V\), whether \(V\) is split is equivalent to whether \(\Pi(D)\) is semisimple.NEWLINENEWLINENEWLINEAs mentioned in section 8 of the paper, the best way to test the definition of \(\Pi(D)\) is to check its appearance in the cohomology of a Shimura curve. The author emphasizes that this is a non-trivial but still seemingly manageable question. A recent work of Yiwen Ding proved that the second piece of the socle filtration on \(\Pi(D)\) appears in the cohomology of a Shimura curve, extending a prior work of the author and \textit{M. Emerton} [Astérisque 331, 255--315 (2010; Zbl 1251.11043)]. This result greatly supports the expectation of the author.NEWLINENEWLINEThis paper is elegantly written and explains the basic tools in this area very well.NEWLINENEWLINEFor the entire collection see [Zbl 1237.11001].