We use the endoscopic classification of automorphic representations of even-dimensional unitary groups to construct level-raising congruences.

We show how the modular representation theory of inner forms of general linear groups over a non-Archimedean local field can be brought to bear on the complex theory in a remarkable way. Let $\text{F}$ be a non-Archimedean locally compact field of residue characteristic $p$, and let $\text{G}$ be an inner form of the general linear group $\text{GL}_{n}(\text{F})$ for $n\geqslant 1$. We consider the problem of describing explicitly the local Jacquet–Langlands correspondence $\unicode[STIX]{x1D70B}\mapsto _{\text{JL}}\unicode[STIX]{x1D70B}$ between the complex discrete series representations of $\text{G}$ and $\text{GL}_{n}(\text{F})$, in terms of type theory. We show that the congruence properties of the local Jacquet–Langlands correspondence exhibited by A. Mínguez and the first author give information about the explicit description of this correspondence. We prove that the problem of the invariance of the endo-class by the Jacquet–Langlands correspondence can be reduced to the case where the representations $\unicode[STIX]{x1D70B}$ and $_{\text{JL}}\unicode[STIX]{x1D70B}$ are both cuspidal with torsion number $1$. We also give an explicit description of the Jacquet–Langlands correspondence for all essentially tame discrete series representations of $\text{G}$, up to an unramified twist, in terms of admissible pairs, generalizing previous results by Bushnell and Henniart. In positive depth, our results are the first beyond the case where $\unicode[STIX]{x1D70B}$ and $_{\text{JL}}\unicode[STIX]{x1D70B}$ are both cuspidal.

We use the method of Ash and Stevens to prove the existence of small slope $p$-adic families of cohomological modular forms for an indefinite quaternion algebra $B$. We prove that the Jacquet–Langlands correspondence relating modular forms on $\text{G}{{\text{L}}_{\text{2}}}/\mathbb{Q}$ and cohomomological modular forms for $B$ is compatible with the formation of $p$-adic families. This result is an analogue of a theorem of Chenevier concerning definite quaternion algebras.

In a paper by Badulescu [Global Jacquet–Langlands correspondence, multiplicity one and classification of automorphic representations, Invent. Math. 172 (2008), 383–438], results on the global Jacquet–Langlands correspondence, (weak and strong) multiplicity-one theorems and the classification of automorphic representations for inner forms of the general linear group over a number field were established, under the assumption that the local inner forms are split at archimedean places. In this paper, we extend the main local results of that article to archimedean places so that the above condition can be removed. Along the way, we collect several results about the unitary dual of general linear groups over ℝ, ℂ or ℍ which are of independent interest.