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Semi-stable representations of \(\text{GL}_2(\mathbb Q_p)\)-adic half-plane and modulo \(p\) reduction. (Représentations semi-stables de \(\text{GL}_2(\mathbb Q_p)\), demi-plan \(p\)-adique et réduction modulo \(p\).) (French. English summary) Zbl 1271.11106
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, 117-178 (2010).
Let \(L\) be a finite extension of \(\mathbb Q_p\), with \(\mathcal O\) the ring of integers of \(L\) and let \(\pi\) be a uniformizer of \(\mathcal O\). Let \(\mathbb F:= \mathcal O/(\pi)\). Let \(k\) be an even integer such that \(4\leq k\leq p+1\). Let \(G:=\text{GL}_2(\mathbb Q_p)\), and let \(\text{St}_G\) denote the Steinberg representation of \(G\). Let \(B(k,{\mathcal L})\) denote a certain completion of \[ |\text{det}|^{{k\over 2}-1}\otimes \text{Sym}^{k-2}\otimes \text{St}_G \] with respect to \(G\)-invariant \(\mathcal O\)-lattices, depending on a parameter \({\mathcal L}\in L\) (defined by C. Breuil [Ann. Sci. Éc. Norm. Supér. (4) 37, No. 4, 559–610 (2004; Zbl 1166.11331)]). The reduction modulo \(\pi\) of \(B(k,{\mathcal L})\) is related to the reduction modulo \(\pi\) of non-crystalline \(p\)-adic representations of \(\text{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)\).
Let \({\mathcal H}\) denote the formal scheme of the \(p\)-adic upper half-plane. The authors define a certain sheaf \(\omega(k,{\mathcal L})\) of \(\mathcal O\)-modules, which is an extension of a sheaf of free \(\mathcal O\)-modules of finite type by a coherent sheaf. It turns out that \(B(k,{\mathcal L})^*\) after being twisted in an appropriate way, becomes \(G\)-isomorphic to \(H^0/{\mathcal H},\omega(k,{\mathcal L}))\otimes L\). The authors calculate \(H^0({\mathcal H},\omega(k,{\mathcal L}))\otimes \mathbb F\) when \(\text{val}({\mathcal L})\geq 0\) (extending the calculations done by J. T. Teitelbaum [Invent. Math. 113, No. 3, 561–580 (1993; Zbl 0806.11027)].
For the entire collection see [Zbl 1192.11002].
11S37 Langlands-Weil conjectures, nonabelian class field theory
22E50 Representations of Lie and linear algebraic groups over local fields
11F80 Galois representations
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