The Jacquet-Langlands correspondence links together (isomorphism classes of) smooth irreducible representations of a general linear group over a \(p\)-adic field \(F\), with coefficients in an algebraic closed field \(\Omega\) of characteristic 0 and smooth irreducible \(\Omega\)-representations of one of its inner forms, that is a general linear group over a central division algebra \(D\) of finite dimension \(d^2\) over \(F\) (\(d\geq 2\)). It can be expressed in two ways (\(m\in \mathbb{N}^*\)):

(i)

a bijection \(\pi_{m,\Omega}\) between the sets \(\mathcal{D}(\mathrm{GL}_{m}(D),\Omega)\) and \(\mathcal{D}(\mathrm{GL}_{md}(F),\Omega)\) of irreducible essentially square integrable \(\Omega\)-representations of \(\mathrm{GL} _{m}(D)\) and \(\mathrm{GL} _{md}(F)\) respectively, characterized by a character identity;

(ii)

a homomorphism \(J_{m,\Omega}\), from the Grothendieck group of the category of smooth finite-length \(\Omega\)-representations of \(\mathrm{GL} _{md}(F)\), \(\mathcal{R}(\mathrm{GL} _{md}(F),\Omega)\), to that of \(\mathrm{GL} _{m}(D)\), \(\mathcal{R}(\mathrm{GL} _{m}(D),\Omega)\), whose restriction to \(\mathcal{D}(\mathrm{GL} _{md}(F),\Omega)\) is the inverse bijection of \(\pi_{m,\Omega}\).

When \(\Omega\) is an algebraic closure \(\overline {\mathbb{Q}}_{\ell}\) of \(\mathbb{Q}_{\ell}\) (\(\ell \neq p\)), we distinguish the set of integral representations over which a reduction modulo \(\ell\) map is defined with image in the set of \(\ell\)-modular representations. The purpose of this paper is to prove that the Jacquet-Langlands correspondence can be reduced modulo \(\ell\), more precisely, that if two integral representations have the same reduction modulo \(\ell\) then so are their images under \(\pi_{m,\Omega}\) or \(J_{m,\Omega}\) (Theorems 1.1 and 1.16 resp.).
In his paper [Proc. Lond. Math. Soc. 104, No 4, 690–727 (2012; Zbl 1247.14025)], J.-F. Dat investigated this question in the case \(m=1\) in order to define a modular Jacquet-Langlands correspondence. The outline of his proof and the theory of Brauer characters that he develops are also found in the proof of the first theorem.
This proof can be divided into three steps. First, the authors reduce the general case to the case of a correspondence between essentially square integrable representations of \(G=\mathrm{GL} _{m}(D)\) and the ones of the multiplicative group \(A\) of a central division algebra of dimension \((md)^2\) over \(F\). Secondly, they extend this correspondence into a group homomorphism \(\mathbf{J}_{\overline {\mathbb{Q}}_{\ell}}\) from the Grothendieck group \(\mathcal{R}(G,\overline {\mathbb{Q}}_{\ell})\) to the one of \(A\) and, following the same line of reasoning as Dat [loc. cit.], they define an analog \(\mathbf{J}_{\overline {\mathbb{F}}_{\ell}}\) from \(\mathcal{R}(G,\overline {\mathbb{F}}_{\ell})\) to \(\mathcal{R}(A,\overline {\mathbb{F}}_{\ell})\) which is compatible to \(\mathbf{J}_{\overline {\mathbb{Q}}_{\ell}}\) by reduction modulo \(\ell\). The third and awkward step is the study of the image under the reduction modulo \(\ell\) map of the set of integral Speh \(\overline {\mathbb{Q}}_{\ell}\)-representations.
With this aim in mind, the authors develop an original argument to count the classes of \(\ell\)-modular representations which leave in the image under the reduction modulo \(\ell\) map of the set of inertial classes of integral Speh representations based on the definition of a new invariant attached to an integral Speh representation and preserved by \(\mathbf{J}_{\overline {\mathbb{Q}}_{\ell}}\), on an extension of the numerical criterium of Dat [loc. cit.] and on the theory of types. While doing this, they prove a compatibility by reduction modulo \(\ell\) property of the Zelevinski classification [A. Minguez and V. Sécherre, Duke Math. J. 163, No 4, 795–885 (2014; Zbl 1293.22005)] (Proposition 8.7) as a consequence of an explicit calculation of the reduction modulo \(\ell\) of an integral Speh representation. They prove also that the image by \(\mathbf{J}_{\overline {\mathbb{F}}_{\ell}}\) of the reduction modulo \(\ell\) of an integral Speh representation is an effective virtual representation up to a sign and deduce as a corollary a bijection between the set of super-Speh \(\overline {\mathbb{F}}_{\ell}\)- representations of \(G\) (that is, the ones whose cuspidal support is supercuspidal) and the set of irreducible \(\overline {\mathbb{F}}_{\ell}\)- representations of \(A\).
This last result is the starting point of the proof of the second theorem. In this proof, the authors introduce the rings \(\mathcal{R}(R,\overline {\mathbb{Q}}_{\ell})\), direct sum of all \(\mathcal{R}(\mathrm{GL} _{m}(R),\overline {\mathbb{Q}}_{\ell})\), \(m\in \mathbb{N}\) (with \(\mathcal{R}(\mathrm{GL} _{0}(R),\overline {\mathbb{Q}}_{\ell})=\mathbb{Z}\)), where \(R\) denotes \(F\) or \(D\). Then they consider the unique ring homomorphism \(\widetilde {\mathbf{B}}_{\ell}\) from \(\mathcal{R}(F,\overline {\mathbb{Q}}_{\ell})\) to \(\mathcal{R}(D,\overline {\mathbb{Q}}_{\ell})\) induced by the set of the homomorphisms \(J_{m,\overline {\mathbb{Q}}_{\ell}}\), \(m\in \mathbb{N}^*\), as defined by A. I. Badulescu [J. Inst. Math. Jussieu 6, No 3, 349–379 (2007; Zbl 1159.22005)]. Using the last mentioned corollary, the authors define a homomorphism from the commutative \(\mathbb{Z}\)-algebra \(\mathcal{R}(F,\overline {\mathbb{F}}_{\ell})\) generated by the set of super-Speh \(\overline {\mathbb{F}}_{\ell}\)-representations of \(\mathrm{GL} _{m}(F)\) for all \(m\in \mathbb{N}^*\), to the “similar” commutative \(\mathbb{Z}\)-algebra \(\mathcal{R}(D,\overline {\mathbb{F}}_{\ell})\) which is compatible to \(\widetilde {\mathbf{B}}_{\ell}\) by reduction modulo \(\ell\) on the set of all integral representations in \(\mathcal{R}(F,\overline {\mathbb{Q}}_{\ell})\) whose reduction modulo \(\ell\) are super-Speh. Next, they prove that this property extends to the set of all integral Speh representations in \(\mathcal{R}(F,\overline {\mathbb{Q}}_{\ell})\) by writing down the reduction modulo \(\ell\) of such a representation in terms of super-Speh \(\overline {\mathbb{F}}_{\ell}\)-representations.