Special \(p\)-adic series and completed étale cohomology. (Série spéciale \(p\)-adique et cohomologie étale complétée.) (French. English summary) Zbl 1246.11106

Breuil, Christophe

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

Keywords:

\(p\)-adic special series; \(p\)-adic Langlands correspondence; completed étale cohomology