## 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].

### MSC:

 11S37 Langlands-Weil conjectures, nonabelian class field theory 22E50 Representations of Lie and linear algebraic groups over local fields 11F80 Galois representations

### Citations:

Zbl 1166.11331; Zbl 0806.11027
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