Représentations cristallines irréductibles de ${\rm GL}_2(\mathbf{Q}_p)$

Details Représentations cristallines irréductibles de ${\rm GL}_2(\mathbf{Q}_p)$ Représentations cristallines irréductibles de ${\rm GL}_2(\mathbf{Q}_p)$

2004-10-04

arxiv.org/abs/math/0410053v1

Mathematics

Number Theory

44 pages, in french

Scientific paper

In \cite[\S1.3]{Br2}, some unitary representations of ${\rm GL}_2(\mathbf{Q}_p)$ on $p$-adic Banach spaces are associated to 2-dimensional irreducible crystalline representations of ${\rm Gal}(\bar{\mathbf{Q}}_p)/\mathbf{Q}_p)$. Some conjectures are formulated concerning those Banach spaces (non triviality, topological irreducibility, admissibility). We prove those conjectures by reinterpreting those Banach as spaces of functions of a certain type on $\mathbf{Q}_p$, and then by using the theory of $(\phi,\Gamma)$-modules associated to the corresponding crystalline representations.

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