The local Jacquet-Langlands correspondence via Fourier analysis

The local Jacquet-Langlands correspondence via Fourier analysis

Let F be a locally compact non-Archimedean field, and let B/F be a division algebra of dimension 4. The JacquetLanglands correspondence provides a bijection between smooth irreducible representations π′ of B× of dimension > 1 and irreducible cuspidal representations of GL2(F ). We present a new construction of this bijection in which the preservation of epsilon factors is automatic. This is don...

The Jacquet-Langlands correspondence via `-adic uniformization

THE JACQUET-LANGLANDS CORRESPONDENCE FOR GL2 Contents

1.1. Matching regular semisimple conjugacy classes. Let X(F ) = F × F×, viewed as the space of semisimple conjugacy classes in GL2(F ) via GL2(F ) → X(F ) sending γ 7→ (Tr(γ),det(γ)). Similarly we have D× → X(F ) sending γ′ 7→ (TrD/F (γ),NmD/F (γ′)), where TrD/F and NmD/F mean reduced trace and norm. For regular semisimple γ ∈ GL2(F ) and γ′ ∈ D×, we write γ ∼ γ′ if they have the same image in ...

Some Endoscopic Properties of The Essentially Tame Jacquet-Langlands Correspondence

Let F be a non-Archimedean local field of characteristic 0 and G be an inner form of the general linear group G = GLn over F . We show that the rectifying character appearing in the essentially tame Jacquet-Langlands correspondence of Bushnell and Henniart for G and G can be factorized into a product of some special characters, called zeta-data in this paper, in the theory of endoscopy of Langl...

On the Jacquet–Langlands correspondence for Hilbert modular forms

2 Modular forms and Automorphic forms 2 2.1 Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 The Adele group of GL2 over Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Automorphic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Hecke act...