Jacquet–Langlands correspondence for symplectic similitude groups
Let be the space of Hilbert–Siegel cusp forms of weight and level on the split symplectic similitude group, let be the corresponding space of algebraic Hilbert–Siegel cusp forms on the quaternionic inner form, and let and be their Hecke algebras away from . Jacquet–Langlands correspondence. The Hecke algebras and are isomorphic, and there is a compatible isomorphism of Hecke modules
This conjecture predicts that the Hecke-module structure of the Hilbert–Siegel cusp forms on the split group can be computed through the quaternionic inner form, providing the proposed Jacquet–Langlands transfer for symplectic similitude groups. The supplied text does not state whether the correspondence has been proved or disproved.
References
Primary source
Clifton Cunningham and Lassina Dembele, “Computing genus 2 Hilbert-Siegel modular forms over (5) via the Jacquet-Langlands correspondence”, arXiv:0704.0011 (2008).
Progress summary
A theorem handles important special cases, but the full correspondence between the two kinds of cusp forms remains unproved.
The conjecture asserts that the Hecke algebras and Hecke modules for the split and quaternionic inner forms agree. The formulation was discussed for by Ihara in 1964; the unrestricted statement is not established by the retrieved sources.
Known results
- Ibukiyama (1984) gave numerical evidence over .
- Sorensen (2008) established a correspondence when is even.
- Computations matched selected Euler factors but explicitly did not prove the correspondence.
2019 restricted-case theorem
A 2019 paper claims a geometric theorem giving an injective prime-to- Hecke-equivariant map for -new forms at squarefree level, with partial image identification. Its scope does not clearly cover the stated arbitrary-weight, arbitrary-level correspondence, so this is claimed progress rather than a verified solution.
Current status (as of August 2026): the even-degree case and other restricted cases are reported as established or claimed, but the full arbitrary-field, arbitrary-weight, arbitrary-level correspondence remains open and unverified.
Sources
- arxiv.org
- ar5iv.labs.arxiv.org
- arxiv.org
- ar5iv.labs.arxiv.org
- arxiv.org
- arxiv.org
- arxiv.org
- pdfs.semanticscholar.org
- arxiv.org
- webusers.imj-prg.fr
- quantamagazine.org
- arxiv.org
- arxiv.org
- ar5iv.labs.arxiv.org
- mathstodon.xyz
- quantamagazine.org
- quantamagazine.org
- scientificamerican.com
- quantamagazine.org
- arxiv.org
- arxiv.org
- arxiv.org
- arxiv.org
- mathstodon.xyz
- mathstodon.xyz
- quantamagazine.org
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