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Special $$p$$-adic series and completed étale cohomology. (Série spéciale $$p$$-adique et cohomologie étale complétée.) (French. English summary) Zbl 1246.11106
Berger, Laurent (ed.) et al., Représentations $$p$$-adiques de groupes $$p$$-adiques III: Méthodes globales et géométriques. Paris: Société Mathématique de France (ISBN 978-2-85629-282-2/pbk). Astérisque 331, 65-115 (2010).
Summary: Let $$f$$ be a new modular parabolic form of weight $$k\geq 2$$ on $$\Gamma_0(Np)$$ with the eigenvector of Hecke operators $$(N, p)=1$$. Let $$E$$ be a finite extension of $$\mathbb Q$$ containing the eigenvalues. If $$k>2$$, we show that the closure of the representation $$\text{Sym}^{k-2}E^2\otimes\pi_p(f)$$ of $$\text{GL}_2(\mathbb Q)$$ in the $$p$$-adic completion $$\varprojlim_n\varinjlim_r H^1(Y(Np^r), \mathbb Z/p^n\mathbb Z)\otimes E$$ gives the invariant $$\mathcal L$$ of $$f$$, that is the restriction of the $$p$$-adic Galois representation of $$f$$ to $$\text{Gal}(\overline{\mathbb Q}/\mathbb Q)$$. By using Colmez’s results we give an explicit description of this closure. The case $$k=2$$ behaves differently, but we show how one can still find the invariant $$\mathcal L$$ from the $$\text{GL}_2(\mathbb Q)$$ viewpoint, in the previous $$p$$-adic completion.
For the entire collection see [Zbl 1192.11002].

##### MSC:
 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11F80 Galois representations
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