Jacquet modules and the Langlands classification

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 ...

On Jacquet–Langlands isogeny over function fields

Article history: Received 15 August 2010 Revised 31 December 2010 Accepted 3 January 2011 Available online xxxx Communicated by David Goss MSC: primary 11G18, 11G09 secondary 14H40

The Eisenstein Ideal and Jacquet-Langlands Isogeny over Function Fields

Let p and q be two distinct prime ideals of Fq[T ]. We use the Eisenstein ideal of the Hecke algebra of the Drinfeld modular curveX0(pq) to compare the rational torsion subgroup of the Jacobian J0(pq) with its subgroup generated by the cuspidal divisors, and to produce explicit examples of Jacquet-Langlands isogenies. Our results are stronger than what is currently known about the analogues of ...

Jacquet modules and induced representations

We discuss some application of Jacquet modules in the study of parabolically induced representations of classical p-adic groups. The main problem that we consider is the construction of non-cuspidal square integrable representations. Since the problem is related to understanding of the reducibilityof the parabolically induced representations, we discuss also the reducibility questions.