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The Restricted Geometric Langlands Equivalence in Positive Characteristic

OpenAI · September 24, 2026

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Abstract

We prove the $\overline{\mathbb Q}_\ell$-linear restricted geometric Langlands equivalence for smooth projective connected curves and connected reductive groups in characteristic p > 0, with ℓ ≠ p, in two regimes. Over $\overline{\mathbb F}_q$, we assume the four characteristic conditions of the restricted theory stated below. Over arbitrary algebraically closed fields, we assume these conditions and additionally that p is very good and does not divide the Weyl-group order. This proves Gaitsgory–Raskin's full-support conjecture [[12, Conjecture 1.3.10]](https://arxiv.org/abs/2508.02237v1) in these regimes.

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Cite

@misc{OAI:The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic-September-24-2026,
  author = {{OpenAI}},
  title = {{The Restricted Geometric Langlands Equivalence in Positive Characteristic}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic-September-24-2026/paper.pdf}{OAI:The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic-September-24-2026}},
  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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