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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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##### References:
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