Approche non-invariante de la correspondance de Jacquet-Langlands : analyse spectrale

TL;DR

This paper completes the proof of the global Jacquet-Langlands correspondence using non-invariant trace formulas, focusing on spectral transfer and its arithmetic implications, extending key theorems in automorphic forms.

Contribution

It introduces the notion of non-invariant spectral transfer and proves its equivalence with geometric transfer, advancing the understanding of trace formulas and automorphic representations.

Findings

Complete proof of the global Jacquet-Langlands correspondence.

Establishment of the equivalence between spectral and geometric transfer.

Extension of Kazhdan's theorem within the non-invariant trace formula framework.

Abstract

This is the second article in a two-part series presenting a new proof comparing the non-invariant trace formula for a general linear group with that of one of its inner forms. In this article, we focus on the spectral side of the trace formula. We complete the proof of the global Jacquet-Langlands correspondence using the non-invariant trace formula and examine its arithmetic implications. Furthermore, we define the notion of non-invariant spectral transfer of a test function and show that it coincides with the non-invariant geometric transfer introduced in our first article. This provides a positive answer to a conjecture of Arthur and extends a well-known theorem of Kazhdan within our framework.