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Browse › Number theory › 014 Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan

Frobenius Structures on Tame Hecke Eigensheaves

OpenAI · October 5, 2026

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Abstract

We construct nonzero perverse Weil Hecke eigensheaves for SLn with Borel level at one marked point, after a finite extension of the field of constants. The parameter is a geometrically dense arithmetic PGLn-local system with tame regular-unipotent monodromy on a once-punctured curve of genus at least two over a finite field of characteristic p > n. The full multi-leg eigenstructure is Frobenius compatible with the prescribed arithmetic eigenvalue, and the geometric sheaf has parabolic nilpotent singular support.

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Possibly relevant literature (found by the agent, not checked)

  • Shiyu Shen, Tamely ramified geometric Langlands correspondence in positive characteristic (arXiv:1810.12491, v3 April 2024; DOI 10.1093/imrn/rnae005). Proves a tamely ramified $\mathrm{GL}_n$ correspondence on an open spectral locus in characteristic $p>n$ with parabolic level, using crystalline differential operators rather than $\ell$-adic local systems; a different-coefficient antecedent to the $\mathrm{SL}_n$, $p>n$ setting of Theorem 1.1 here, which this paper does not cite.

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Cite

@misc{OAI:Frobenius-Structures-on-Tame-Hecke-Eigensheaves-October-5-2026,
  author = {{OpenAI}},
  title = {{Frobenius Structures on Tame Hecke Eigensheaves}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Frobenius-Structures-on-Tame-Hecke-Eigensheaves-October-5-2026/tame-hecke-frobenius.pdf}{OAI:Frobenius-Structures-on-Tame-Hecke-Eigensheaves-October-5-2026}},
  year = {2026}
}

Family 014: Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan

Proves the restricted geometric Langlands equivalence for connected reductive groups on smooth projective connected curves over $\overline{\mathbb F}_q$ under the four stated Lie-theoretic characteristic hypotheses. Over arbitrary algebraically closed fields of characteristic p > 0, the same conclusion holds assuming additionally that p is very good for the group and $p\nmid |W_G|$. Over global function fields, proves Ramanujan at every place for globally generic cuspidal representations of split adjoint absolutely simple exceptional groups, without characteristic or ramification-depth restrictions, and at every unramified place for cuspidal representations of split adjoint absolutely simple groups with a generic unramified component. Assuming the finite-level Ramanujan–Arthur decomposition, constructs global Arthur enhancements of occurring cuspidal excursion parameters for split connected semisimple groups at full finite level, recovering the given parameters by diagonal specialization on the entire Weil group, including inertia.

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