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- W1573244699 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 $(0,k-1)(kin mathbf {Z},1<k<p)$ 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 $Gammasb 0(pN)(pnmid N)$ 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." @default.
- W1573244699 created "2016-06-24" @default.
- W1573244699 creator A5029078442 @default.
- W1573244699 creator A5071464327 @default.
- W1573244699 date "2002-11-01" @default.
- W1573244699 modified "2023-10-14" @default.
- W1573244699 title "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 ${rm GL}sb 2$" @default.
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- W1573244699 doi "https://doi.org/10.1215/s0012-7094-02-11522-1" @default.
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