Geodesic
Hodge classes and the Jacquet–Langlands correspondence
Forum of Mathematics, Pi, Tome 11 (2023)

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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 $. 
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Atsushi Ichino; Kartik Prasanna. Hodge classes and the Jacquet–Langlands correspondence. Forum of Mathematics, Pi, Tome 11 (2023). doi: 10.1017/fmp.2023.20

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