Multiplicités modulaires et représentations de ${\rm GL}\sb 2(\mathbf {Z}\sb p)$ et de ${\rm Gal}(\overline {\mathbf {Q}}\sb p/\mathbf {Q}\sb p)$ en $\ell=p$. Appendice par Guy Henniart. Sur l'unicité des types pour ${\rm GL}\sb 2$

Abstract

We formulate a conjecture giving a link between the various rings parametrizing the $2$-dimensional potentially semistable $p$-adic representations of ${\rm Gal}(\overline {\mathbf {Q}}\sb p/\mathbf {Q}\sb p)$ with Hodge-Tate weights $(0,k-1)(k\in \mathbf {Z},1<k<p)$ having the same reduction modulo $p$ and the representations of ${\rm GL}\sb 2(\mathbf {Z}\sb p)$ that are used, via compact induction, to build the smooth irreducible representations of ${\rm GL}\sb 2(\mathbf {Q}\sb p)$. We prove this conjecture for semistable representations and $k$ even. In doing this, we obtain precise results on the restriction to ${\rm Gal}(\overline {\mathbf {Q}}\sb p/\mathbf {Q}\sb p)$ of the representations of ${\rm Gal}(\overline {\mathbf {Q}}/\mathbf {Q})$ over $\overline {\mathbf {F}}\sb p$ that are associated to modular forms on $\Gamma\sb 0(pN)(p\nmid N)$ of weight smaller than $p$. In an appendix, G. Henniart determines which smooth irreducible representations $\lambda$ of ${\rm GL}\sb 2(\mathfrak {O}\sb F)$ are typical for ${\rm GL}\sb 2(F)$ (where $F$ is a locally compact non-Archimedean field and $\mathfrak {O}\sb F$ its ring of integers), in the sense that there is a component $s(\lambda)$ in the Bernstein decomposition (for the category of smooth representations of ${\rm GL}\sb 2(F)$) such that $\lambda$ appears only in smooth representations of ${\rm GL}\sb 2(F)$ belonging to $s(\lambda)$. We need Henniart's results, in the case $F=\mathbf {Q}\sb p$, to state the aforementioned conjecture.