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The Trace Conjecture for derived Hitchin stacks

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.

Sources & referencesView supporting material

Primary source

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

derived-geometryhitchin-systemsshtuka-spaces

Progress summary

Refreshed
Partially solved

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.

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Solutions 0

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