\(p\)-adic-representations of \(\text{GL}_2(L)\) and derived categories
The article concerns the interesting correspondence between \(p\)-adic two-dimensional representations \(\rho\) of the Galois group \(G_p\) of \(\mathbb{Q}_p\), and unitary representations \(\pi\) of \(\text{GL}(2,\mathbb Q_p)\) on \(p\)-adic Banach spaces. Breuil associated to a semistable non-cristalline two-dimensional \(\rho\) a locally analytic \(\pi\). An invariant \(\mathfrak{L}\) appears. It does not appear in the ``classical'' (complex) Langlands correspondence. The unitary completion of the \(\pi\) is admissible and nonzero, and its mod \(p\) reduction is compatible with the mod \(p\) correspondence defined by Breuil. The main result of this paper is that on considering the space of morphisms, in a suitable derived category, between the dual of \(\pi\) and the de Rham complex of Drinfeld's half-plane -- a rank two admissible filtered \((\varphi,N)\)-module -- one finds the Fontaine module associated with the original \(\rho\). The work extends from \(\mathbb Q_p\) to a finite extension \(L\). Then the locally \(\mathbb Q_p\)-analytic representations \(\pi\) of \(\text{GL}(2,L)\) become parametrized not by a single \(\mathfrak{L}\), but by \((\mathfrak{L}_\sigma)\), where \(\sigma\) ranges over the embeddings of \(L\) in the algebraic closure of \(\mathbb Q_p\). The author explains that the \(\pi\) obtained might not be the best objects when \(L\not=\mathbb{Q}_p\). For example, it is not clear that their unitary completion is admissible. But he hopes they are subobjects of the correct representations.
\(p\)-adic representations of GL(2) over a finite extension of \(\mathbb Q_p\)
scientific article; zbMATH DE number 5728352
scientific article