Hodge classes and the Jacquet-Langlands correspondence

Abstract

We prove that the Jacquet-Langlands correspondence for cohomological automorphic forms on quaternionic Shimura varieties is realized by a Hodge class. Conditional on Kottwitz's conjecture for Shimura varieties attached to unitary similitude groups, we also show that the image of this Hodge class in $\ell$-adic cohomology is Galois invariant for all $\ell$.

118 pages. Revised version following referee reports. Changes to introduction, Sec. 9 and Sec. 11, minor edits elsewhere. To be published in Forum of Math, Pi

References in corpus (2)

• Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups
• On Tate conjecture for the special fibers of some unitary Shimura varieties

Cited by in corpus (2)

• The endoscopic fundamental lemma for unitary Friedberg-Jacquet periods
• Endoscopy for unitary symmetric spaces