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Geometric Jacquet–Langlands correspondence for Shimura varieties

Let (G,X)(G,X) and (G′,X′)(G',X') be Shimura data whose underlying groups are pure inner forms of one another, and let Ig⁡(G,X)\operatorname{Ig}(G,X) and Ig⁡(G′,X′)\operatorname{Ig}(G',X') denote the associated Igusa stacks. The Igusa-stack conjecture predicts the existence of exotic isomorphisms Ig⁡(G,X)≅Ig⁡(G′,X′)\operatorname{Ig}(G,X)\cong \operatorname{Ig}(G',X') relating the two Shimura data, in a manner that yields—via the Fargues–Scholze spectral action—a geometric realization of the Jacquet–Langlands correspondence between the cohomology of the associated Shimura varieties.

References

Primary source

arXiv

Additional references

arithmetic-geometryautomorphic-formsshimura-varieties
Posted · edited

Progress summary

Refreshed
Claimed progress

A September preprint reports the correspondence in many cases, but the general problem remains open.

The problem asks for a geometric realization of Jacquet–Langlands correspondence through isomorphisms of Igusa stacks attached to pure inner forms. Pol van Hoften and Jack Sempliner formulated this conjectural framework and reported substantial cases in a preprint submitted on September 8, 2026.

Known results

  • A 2004 result realizes certain unitary-group Jacquet–Langlands correspondences in Shimura-variety cohomology.
  • A 2023 work treats substantial PEL cases of type AC under unramifiedness and related hypotheses, while leaving functoriality and compactification issues open.
  • A December 2025 paper proves the Igusa-stack fiber-product conjecture for meta-unitary Shimura varieties and derives cohomological consequences.
  • March and April 2026 papers develop further Igusa-stack cases and a locally analytic Jacquet–Langlands correspondence.

September 8, 2026 preprint

Van Hoften and Sempliner state that they prove the Igusa-stack conjecture in many cases of interest and use the Fargues–Scholze spectral action to obtain cohomological consequences. This is a claimed advance in a new preprint, not a complete proof in general, and no independent verification or objection was found.

Current status (as of September 2026): special cases and related constructions are reported, while the full geometric Jacquet–Langlands correspondence remains open and the latest broad claim is unverified.

Sources

Solutions 0

No solutions have been posted yet.