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On the reduction modulo $$p$$ of representations of a quaternion division algebra over a $$p$$-adic field. (English) Zbl 1321.11117
The local Langlands correspondence for $$\mathrm{GL}_2$$ over a non-archimedean local field $$F$$ is a bijection between the (isomorphism classes of) two-dimensional Frobenius semi-simple Deligne representations of the Weil group $$W_F$$ of $$F$$ and the (isomorphism classes of) irreducible admissible representations of $$\mathrm{GL}_2(F)$$. On the other hand, there is also the $$p$$-adic and mod $$p$$ Langlands correspondence for $$\mathrm{GL}_2(F)$$. The $$p$$-adic correspondence is concerned with a parametrization of $$p$$-adic Galois representations, i.e., continuous representations of the absolute Galois group $$G_F$$ of $$F$$ in a vector space over a $$p$$-adic field. The mod $$p$$ correspondence is the Langlands correspondence between representations of $$\mathrm{GL}_2$$ and the absolute Galois group $$G_F$$ in vector spaces over the algebraic closure $$\overline{\mathbb{F}}_p$$ of the finite field $$\mathbb{F}_p$$. One of the key ideas in the study of these correspondences is to verify whether they are compatible with reduction modulo $$p$$.
In the case of $$\mathrm{GL}_2$$ over the field $$F=\mathbb{Q}_p$$, these correspondences were studied by C. Breuil [Compos. Math. 138, No. 2, 165–188 (2003; Zbl 1044.11041); J. Inst. Math. Jussieu 2, No. 1, 23–58 (2003; Zbl 1165.11319)]. It turns out that they are compatible with the reduction modulo $$p$$. The point of the present paper is to study the same problem for the multiplicative group $$D^\times$$ of the quaternion division algebra $$D$$ over $$F$$.
The local Jacquet-Langlands correspondence [H. Jacquet and R. P. Langlands, Automorphic forms on $$\mathrm{GL}(2)$$. Lecture Notes in Mathematics, 114. Berlin-Heidelberg-New York: Springer-Verlag (1970; Zbl 0236.12010)] provides a correspondence between the (isomorphism classes of) discrete series representations of $$\mathrm{GL}_2(F)$$ and the (isomorphism classes of) irreducible admissible representations of $$D^\times$$. Combined with the local Langlands correspondence, this gives a parametrization of irreducible admissible representations of $$D^\times$$ in terms of representations of $$W_F$$. The problem is to find out whether this correspondence is compatible with the semisimplification of the reduction modulo $$p$$. To answer this question, the author provides an explicit classification of mod $$p$$ irreducible representations of $$D^\times$$ and certain irreducible representations of $$W_F$$. Such classification implies the mod $$p$$ correspondence. Computing the reduction modulo $$p$$ shows that the local Langlands correspondence composed with the Jacquet-Langlands correspondence is not compatible with the mod $$p$$ correspondence, except in the case of level zero, which was already proved by M.-F. Vignéras [Lect. Notes Math. 1380, 254–266 (1989; Zbl 0694.12012)].

##### MSC:
 11S37 Langlands-Weil conjectures, nonabelian class field theory 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11F80 Galois representations 20C20 Modular representations and characters
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##### References:
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