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Modular multiplicities and representations of \(\text{GL}_2(\mathbb Z_p)\) and \(\text{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)\) at \(\ell=p\). (Multiplicités modulaires et représentations de \(\text{GL}_2(\mathbb Z_p)\) et de \(\text{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)\) en \(\ell=p\).) (French) Zbl 1042.11030
From the authors’ abstract: We formulate a conjecture giving a link between the various rings parameterizing the 2-dimensional potentially semistable \(p\)-adic representations of \(\text{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)\) with Hode-Tate weights \((0,k-1)\) \((k\in\mathbb Z\), \(1< k< p\)) having the same reduction modulo \(p\) and the representations of \(\text{GL}_2(\mathbb Z_p)\) that are used, via compact induction, to build the smooth irreducible representations of \(\text{GL}_2(\mathbb Q_p)\). We prove this conjecture for semistable representations and \(k\) even. In doing this, we obtain precise results on the restriction to \(\text{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)\) of the representations of \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\) over \(\overline{\mathbb F}_p\) that are associated to modular forms on \(\Gamma_0(pN)\) \((p\nmid N)\) of weight smaller than \(p\).

MSC:
11F80 Galois representations
11F85 \(p\)-adic theory, local fields
11S23 Integral representations
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