Approche non-invariante de la correspondance de Jacquet-Langlands : analyse spectrale
Yan-Der Lu
TL;DR
The paper advances the Jacquet–Langlands program by completing a non-invariant trace formula-based proof of the global correspondence, with a deep spectral focus. It introduces and proves a non-invariant spectral transfer of test functions and shows its equivalence to the geometric transfer from the first part, thereby validating Arthur’s conjecture in this GL and inner-form setting. The work also derives arithmetic consequences, isolates spectral components, and extends classic results like Kazhdan’s theorem to the non-invariant framework. Altogether, it provides a robust bridge between geometric transfer identities and spectral transfer, yielding a comprehensive global LJ correspondence and new insights into L- and ε-factors under transfer. The results have potential implications for endoscopy, beyond-endoscopy perspectives, and Ramanujan–Petersson-type phenomena for inner forms of GL.
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.