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The Trace Conjecture for derived Hitchin stacks

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Let M\mathcal M be a derived Hitchin stack, let Sht⁡M\operatorname{Sht}_{\mathcal M} be its associated shtuka stack, and let c1c_1 be the correspondence map whose derived dimension is d(c1)d(c_1). Let cM\mathfrak c_{\mathcal M} be the cohomological correspondence constructed from Frobenius and the diagonal, and let Tr⁡(cM)∈CH⁡d(c1)(Sht⁡M)\operatorname{Tr}(\mathfrak c_{\mathcal M})\in\operatorname{CH}_{d(c_1)}(\operatorname{Sht}_{\mathcal M}) be its trace class. Trace Conjecture. For M\mathcal M a derived Hitchin stack as in the cited work,

Tr⁡(cM)=[Sht⁡M]∈CH⁡d(c1)(Sht⁡M).\operatorname{Tr}(\mathfrak c_{\mathcal M})=[\operatorname{Sht}_{\mathcal M}]\in\operatorname{CH}_{d(c_1)}(\operatorname{Sht}_{\mathcal M}).

The equality is proposed as a necessary ingredient for the corresponding modularity conjecture, although it is not asserted in the full generality of arbitrary derived stacks.

References

Primary source

Tony Feng and Michael Harris, “Derived structures in the Langlands Correspondence”, arXiv:2409.03035 (2025).

derived-geometryhitchin-systemsshtuka-spaces
Posted

Progress summary

Refreshed
Claimed progress

The conjecture has been proved in a restricted low-rank range, while its full range remains open.

Feng and Harris formulate the trace equality as [FH25, Conjecture 9.4.2], as an ingredient for extending modularity beyond the generic fiber. The conjecture concerns identifying the Frobenius–diagonal trace class with the fundamental class of the associated shtuka stack.

August 2026 low-corank proof

A 2026 preprint proves the trace equality for all r≥0r \ge 0 in the range m≤n/3m \le n/3. It then uses an embedding argument to establish the broader modularity conjecture for m≤nm \le n, but does not claim the trace conjecture itself in the remaining range n/3<m≤nn/3 < m \le n.

Current status (as of August 2026): The trace equality is proved for m≤n/3m \le n/3; its validity for n/3<m≤nn/3 < m \le n remains open, even though the associated modularity conjecture is claimed there via an embedding argument.

Sources

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