Let \(F\) be a non-Archimedean locally compact field of residual characteristic \(p\). Let \(D\) be a finite-dimensional central division algebra over \(F\) of reduced degree \(d\). Let \(G = \mathrm{GL}_m(D)\) be the group of invertible elements in the algebra of \(m\times m\) matrices with entries in \(D\), where \(m\geq 1\) is an integer. Let \(H=\mathrm{GL}_n(F)\) be the general linear group, where \(n=md\). Then, \(G\) is an inner form of \(H\). The local Jacquet-Langlands correspondence is a correspondence between (isomorphism classes of) essentially square-integrable irreducible representations of groups \(G\) and \(H\), which is given by a character relation on elliptic regular conjugacy classes [H. Jacquet and R. P. Langlands, Automorphic forms on \(\mathrm{GL}(2)\). Berlin etc.: Springer-Verlag (1970; Zbl 0236.12010); J. D. Rogawski, Duke Math. J. 50, 161–196 (1983; Zbl 0523.22015); P. Deligne et al., in: Représentations des groupes réductifs sur un corps local. Travaux en Cours, 33–117 (1984; Zbl 0583.22009); A. I. Badulescu, Ann. Sci. Éc. Norm. Supér. (4) 35, No. 5, 695–747 (2002; Zbl 1092.11025)].
The Bernstein–Zelevinsky classification [A. V. Zelevinsky, Ann. Sci. Éc. Norm. Supér. (4) 13, 165–210 (1980; Zbl 0441.22014); M. Tadić, J. Reine Angew. Math. 405, 48–77 (1990; Zbl 0684.22008); A. I. Badulescu, J. Inst. Math. Jussieu 6, No. 3, 349–379 (2007; Zbl 1159.22005)] provides a classification of essentially square-integrable irreducible representations in terms of parabolic induction from cuspidal irreducible representations. More precisely, for an essentially square-integrable representation \(\pi\) of \(G\), there exist a unique integer \(r\) dividing \(m\), and a unique (isomorphism class of) cuspidal irreducible representation \(\rho\) of \(\mathrm{GL}_{m/r}(D)\) such that \(\pi\) is isomorphic to the unique irreducible quotient of the parabolically induced representation \[ {\mathrm{Ind}}_{P}^G\left(\rho\otimes\rho\nu^{s(\rho)}\otimes\dots\otimes\rho\nu^{s(\rho)(r-1)}\right), \] where \(P\) is the parabolic subgroup of \(G\) with the Levi factor isomorphic to a product of \(r\) copies of \(\mathrm{GL}_{m/r}(D)\) and \(\nu\) is the absolute value of the reduced norm on \(\mathrm{GL}_{m/r}(D)\). The exponent \(s(\rho)\) is a positive integer attached to \(\rho\) by M. Tadić [J. Reine Angew. Math. 405, 48–77 (1990; Zbl 0684.22008)] as the unique positive integer \(k\) such that the parabolically induced representation \[ {\mathrm{Ind}}_{P'}^{\mathrm{GL}_{2m/r}(D)}\left(\rho\otimes\rho\nu^k\right) \] is reducible, where \(P'\) is the parabolic subgroup of \(\mathrm{GL}_{2m/r}(D)\) with the Levi factor isomorphic to the product of two copies of \(\mathrm{GL}_{m/r}(D)\). Let \(s(\pi)=s(\rho)\), where \(\rho\) is attached to \(\pi\) as above.
This statement of the Bernstein-Zelevinsky classification also captures the case of the group \(H\), by taking \(D=F\) and \(m=n\), and in that case \(s(\sigma)=1\) for any cuspidal irreducible representation \(\sigma\). If \({}_{\mathrm{JL}}\pi\) is the essentially square-integrable representation of \(H\) corresponding to \(\pi\) in the local Jacquet-Langlands correspondence, then there exist a unique positive integer \(u\) dividing \(n\), and a unique (isomorphism class of) cuspidal irreducible representation \(\sigma\) of \(\mathrm{GL}_{n/u}(F)\), such that \({}_{\mathrm{JL}}\pi\) is isomorphic to the unique irreducible quotient of the parabolically induced representation \[ {\mathrm{Ind}}_{Q}^H \left( \sigma\otimes\sigma\nu\otimes\dots\otimes\sigma\nu^{u-1} \right), \] where \(Q\) is the parabolic subgroup of \(H\) with the Levi factor isomorphic to \(u\) copies of \(\mathrm{GL}_{n/u}(F)\). The integers \(r\) and \(u\), attached to \(\pi\) and \({}_{\mathrm{JL}}\pi\) respectively, are related by the equation \[ u=rs(\pi). \] The goal of the explicit local Jacquet–Langlands correspondence is to find the relation between cuspidal irreducible representations \(\rho\) of \(G\) and \(\sigma\) of \(H\), attached to \(\pi\) and \({}_\mathrm{JL}\pi\) by the Bernstein–Zelevinsky classification.
Cuspidal irreducible representations of \(H\) and \(G\) are classified in terms of the theory of simple types [C. J. Bushnell and P. C. Kutzko, The admissible dual of \(\mathrm{GL}(N)\) via compact open subgroups. Princeton, NJ: Princeton University Press (1993; Zbl 0787.22016); P. Broussous, Proc. Lond. Math. Soc., III. Ser. 77, No. 2, 292–326 (1998; Zbl 0912.22007); V. Sécherre, Bull. Soc. Math. Fr. 132, No. 3, 327–396 (2004; Zbl 1079.22016); V. Sécherre, Compos. Math. 141, No. 6, 1531–1550 (2005; Zbl 1082.22011); V. Sécherre, Ann. Sci. Éc. Norm. Supér. (4) 38, No. 6, 951–977 (2005; Zbl 1106.22014); V. Sécherre and S. Stevens, J. Inst. Math. Jussieu 7, No. 3, 527–574 (2008; Zbl 1140.22014)]. A cuspidal irreducible representation \(\rho\) of \(\mathrm{GL}_{m/r}(D)\) is isomorphic to a compactly induced representation from an extended maximal simple type, which is a pair of a compact modulo center subgroup \(J\) of \(\mathrm{GL}_{m/r}(D)\) and an irreducible representation \(\lambda\) of \(J\), and unique up to conjugacy. In view of this, the explicit local Jacquet-Langlands correspondence aims at describing the extended maximal simple type of \(\sigma\) in terms of the extended maximal simple type of \(\rho\).
The previous results regarding the explicit Jacquet-Langlands correspondence are mostly concerned with the cases in which both \(\pi\) and \({}_{\mathrm{JL}}\pi\) are cuspidal. In particular, the case of an essentially tame cuspidal irreducible representation \(\pi\) of \(G\), such that \({}_{\mathrm{JL}}\pi\) is cuspidal, is treated by C. J. Bushnell and G. Henniart [Pure Appl. Math. Q. 7, No. 3, 469–538 (2011; Zbl 1244.11053)], the case of \(n=p^k\), with \(p\neq 2\), and \(\pi\) a cuspidal irreducible representation of the group \(D^\times\) of invertible elements in \(D\), i.e., \(m=1\), which is maximal totally ramified, is treated by the same authors [J. Reine Angew. Math. 580, 39–100 (2005; Zbl 1074.11063)], and the case of an epipelagic cuspidal irreducible representation \(\pi\) of \(G\) is treated by N. Imai and T. Tsushima [Kyoto J. Math. 58, No. 3, 623–638 (2018; Zbl 1446.11097)]. All these results are based on the formula for the trace of \(\pi\) on an elliptic regular element in terms of the trace of the representation \(\lambda\) of \(J\) in the extended maximal simple type of \(\pi\). The case of essentially square-integrable representations of depth zero is treated in [A. J. Silberger and E.-W. Zink, Can. J. Math. 55, No. 2, 353–378 (2003; Zbl 1028.22017); J. Reine Angew. Math. 585, 173–235 (2005; Zbl 1087.22012); C. J. Bushnell and G. Henniart, J. Number Theory 131, No. 2, 309–331 (2011; Zbl 1205.22012)].
The approach towards the explicit local Jacquet-Langlands correspondence of the present paper is very different from the previous work. It exploits the modular representation theory and congruences to relate complex representations. More precisely, let \(\ell\) be a prime different from the residual characteristic \(p\). Fixing an isomorphism between \(\mathbb{C}\) and the algebraic closure \(\overline{\mathbb{Q}}_\ell\) of the field \(\mathbb{Q}_\ell\) of \(\ell\)-adic numbers, gives rise to the local \(\ell\)-adic Jacquet-Langlands correspondence for smooth essentially square-integrable irreducible representations over \(\overline{\mathbb{Q}}_\ell\) of \(G\) and \(H\). According to the work of M.-F. Vignéras [Représentations \(\ell\)-modulaires d’un groupe réductif \(p\)-adique avec \(\ell\neq p\). Boston, MA: Birkhäuser (1996; Zbl 0859.22001); in: Contributions to automorphic forms, geometry, and number theory. Baltimore, MD: Johns Hopkins University Press. Papers from the conference in honor of Joseph Shalika on the occasion of his 60th birthday, Johns Hopkins University, Baltimore, MD, USA, May 14–17, 2002, 773–801 (2004; Zbl 1084.11023)], if a smooth irreducible \(\ell\)-adic representation of \(G\) is integral, one may reduce mod \(\ell\) to obtain a semi-simple representation of finite length over the algebraic closure \(\overline{F}_\ell\) of the finite field \(F_\ell\). Two such \(\ell\)-adic representations of \(G\) are called congruent mod \(\ell\) if their reductions mod \(\ell\) are isomorphic.
The crucial technical ingredient in this modular approach is to show how congruences mod \(\ell\) are related to the theory of types, in particular, the endo-classes of representations. An endo-class is an invariant constructed in terms of the theory of types [C. J. Bushnell and G. Henniart, Publ. Math., Inst. Hautes Étud. Sci. 83, 105–233 (1996; Zbl 0878.11042); P. Broussous et al., Doc. Math. 17, 23–77 (2012; Zbl 1280.22018)], attached to essentially square-integrable irreducible representations of any inner form of any general linear group over \(F\). The Endo-class Invariance Conjecture claims that the endo-classes of \(\pi\) and \({}_{\mathrm{JL}}\pi\) are the same. One of the main results of this paper is the proof of this conjecture under the assumption that the conjecture is true for the case in which \(\pi\) and \({}_{\mathrm{JL}}\pi\) are both cuspidal and of torsion number one. Recall that the torsion number of \(\pi\) is the number of unramified characters \(\chi\) such that \(\pi\chi\) is isomorphic to \(\pi\). The proof is based on the properties of mod \(\ell\) congruences of \(\ell\)-adic representations observed in [J.-F. Dat and M.-F. Vignéras, Proc. Lond. Math. Soc. (3) 104, No. 4, 690–727 (2012; Zbl 1241.22020); A. Mínguez and V. Sécherre, Invent. Math. 208, No. 2, 553–631 (2017; Zbl 1412.22035)]. In fact, two mod \(\ell\) congruent essentially square-integrable irreducible \(\ell\)-adic representations have the same endo-class. Conversely, it is proved in this paper that if two essentially square-integrable irreducible \(\ell\)-adic representations have the same endo-class, then there is a chain of congruences, possibly modulo different primes, which link these two representations. Moreover, if the representations of \(G\) are linked, then the same is true for the representations of \(H\) corresponding to them under the local Jacquet-Langlands correspondence.
The second main result of the paper provides the explicit local Jacquet-Langlands correspondence in the case of essentially tame discrete series representations of \(G\), up to an unramified twist. It is given in terms of the so-called admissible pairs introduced by C. J. Bushnell and G. Henniart [Pure Appl. Math. Q. 7, No. 3, 469–538 (2011; Zbl 1244.11053)], and is a generalization of their results.
After this paper had been written, using [C. J. Bushnell and G. Henniart, Pure Appl. Math. Q. 7, No. 3, 469–538 (2011; Zbl 1244.11053)] and the methods of this paper, A. Dotto [J. Reine Angew. Math. 784, 177–214 (2022; Zbl 1494.22015)] has proved the Endo-class Invariance Conjecture together with the explicit description of the local Jacquet-Langlands conjecture up to inertia.