Let X be a smooth projective curve over an algebraically closed field of characteristic zero, let G be a reductive group, and let check G be its Langlands dual group. Under the geometric Langlands equivalence between D-mod(Bun_G(X)) and the appropriate spectral category on LocSys_check G(X), the constant sheaf on Bun_G(X) corresponds to the spectral object supported at the trivial check G-local system; equivalently, it is the automorphic object associated with the trivial check G-local system.

V. Lafforgue's constant-sheaf conjecture in geometric Langlands

Let $X$ be a smooth projective curve over an algebraically closed field of characteristic zero, let $G$ be a reductive group, and let $\check G$ be its Langlands dual group. Under the geometric Langlands equivalence between $D\text{-}\mathrm{mod}(\mathrm{Bun}_G(X))$ and the appropriate spectral category on $\mathrm{LocSys}_{\check G}(X)$ , the constant sheaf on $\mathrm{Bun}_G(X)$ corresponds to the spectral object supported at the trivial $\check G$ -local system; equivalently, it is the automorphic object associated with the trivial $\check G$ -local system.

References

Primary source

arXiv

Additional references

The image of the constant sheaf under the geometric Langlands equivalence — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

An August 2026 preprint claims to settle this case of geometric Langlands, but the result has not been independently verified.

The problem concerns V. Lafforgue’s conjectural constant-sheaf case of the geometric Langlands correspondence. A recent preprint claims a computation that includes this case and extends to a broader class of local systems.

August 2026 claimed computation

An August 2026 preprint claims to compute the geometric Langlands image of the constant sheaf, thereby confirming the conjectured case and treating broader local systems. The claim remains unrefereed and unverified.

Current status (as of August 2026): The constant-sheaf case is claimed solved by an unrefereed preprint, but independent verification is not recorded.

Sources

No solutions have been posted yet.