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The image of Colmez’s Montreal functor. (English) Zbl 1297.22021

The subject of the paper is the study of \(p\)-adic representations of \(G =\mathrm{GL}_2(\mathbb Q_p)\). Let \(L\) be a finite extension of \((\mathbb Q_p)\), let \(\mathcal O\) be the ring of integers of \(L\) and \(k\) its residue field. Considered are representations of \(G\) on \(L\)-Banach spaces or on \(\mathcal O\) -modules. In a unitary Banach space representation of \(G\) one may choose a \(G\)-invariant \(\mathcal O\)-lattice \(\Theta\) and reduction gives a \(k\)-representation of \(G\). Let \(\mathcal G\) denote the Galois group of \(\overline{\mathbb Q}_p /\mathbb Q_p\).
Colmez’s functor is an exact covariant functor \(\mathbf V\) from the category of smooth \(G\)-representations of finite length with a central character on torsion \(\mathcal O\)-modules to the category of continuous \(\mathcal G\)-representations of finite length on \(\mathcal O\) -modules. The author extends this functor to a functor on \(L\)-Banach space representations. Let \(\Pi\) be an absolutely irreducible admissible unitary \(L\)-Banach representation of \(G\) with a central character. A theorem on the reduction \(\mathrm {mod}-p\) of \(\Pi\), proved here for \(p > 3\), implies that \(\mathrm{dim}_L \mathbf V(\Pi)\leq 2\) and determines for which \(\Pi\) this dimension equals 2. For those \(\Pi\) it is proved that their isomorphism classes correspond bijectively, via the functor \(\mathbf V\), with the absolutely irreducible 2-dimensional continuous \(L\)-representations of \(\mathcal G\) with determinant \(\varepsilon \xi\) (\(\xi\) is the central character of \(\Pi, \varepsilon\) is the cyclotomic character). The final result is an anti-equivalence of categories between the category of admissible unitary \(L\)-Banach representations of \(G\) of finite length, with central character \(\varepsilon\) and all irreducible subquotients isomorphic to \(\Pi\), and the category of modules of finite length over the ring representing the deformation functor of \(V = \mathbf V(\Pi)\) to artinian local \(L\)-algebras.
An essential tool for the proof of these results is a deformation theory for \(k\)-representations of \(G\), which is presented in such a way that it can be used for an arbitrary \(p\)-adic analytic group, when certain hypotheses (on Ext groups) are satisfied.
In the case of \(G =\mathrm{GL}_2(\mathbb Q_p)\) the category of Banach space (resp. \(\mathcal O\)-module) representations of \(G\) can be decomposed into a product of subcategories, indexed by blocks of irreducible smooth \(k\)-representations of \(G\). There are four types of blocks to be considered and for which the application of the theory mentioned above is different. Also results of deformation theory for 2-dimensional representations of \(\mathrm{Gal}(\mathbb Q_p) / (\mathbb Q_p)\) are needed. They are deduced from work of Böckle.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
11S37 Langlands-Weil conjectures, nonabelian class field theory
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