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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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[2] Breuil, C., Sur quelques représentations modulaires et p-adique de \(\operatorname{GL}_2(\mathbb{Q}_p)\) II, J. Inst. Math. Jussieu, 2, 23-58, (2003)
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