View source for Geometric Langlands and the generalized Ramanujan conjecture over function fields

'''Geometric Langlands and the generalized Ramanujan conjecture over function fields''' brings together eight manuscripts in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), dated 24 September and 5 October 2026, on the Langlands program for curves over finite fields and their algebraic closures. They claim the restricted geometric Langlands equivalence in positive characteristic under stated conditions on the characteristic, two conjectures of Gaitsgory, Lafforgue and Raskin on the canonical Arthur filtration and decomposition of automorphic functions, the generalized Ramanujan conjecture for globally generic cuspidal representations of split exceptional groups over global function fields at every place, and constructions of tame Hecke eigensheaves. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== Background ==
* As recounted in the introductions of the manuscripts:
** Over function fields, Drinfeld proved the Ramanujan conjecture for GL<sub>2</sub> and Laurent Lafforgue proved it for GL<sub>''n''</sub>. Lomelí developed the Langlands–Shahidi method over function fields and proved the generic Ramanujan conjecture for split classical groups and quasi-split unitary groups. Vincent Lafforgue's excursion operators attach semisimple global parameters to cuspidal automorphic representations of general reductive groups, and Genestier and Lafforgue proved local–global compatibility for the semisimple local parameters. Sawin and Templier, and Ciubotaru and Harris, proved temperedness at unramified places under additional hypotheses.
** On the geometric side, Arinkin, Gaitsgory, Kazhdan, Raskin, Rozenblyum and Varshavsky introduced the restricted stack of local systems and the spectral action on automorphic sheaves with nilpotent singular support. The characteristic-zero geometric Langlands theorem was established in a five-part project of Arinkin, Beraldo, Campbell, Chen, Faergeman, Gaitsgory, Lin, Raskin and Rozenblyum. In positive characteristic, Gaitsgory and Raskin ([https://arxiv.org/abs/2508.02237 arXiv:2508.02237]) constructed an equivalence onto an open and closed union of spectral components and conjectured that no component is missing (their Conjecture 1.3.10).
** Gaitsgory, Lafforgue and Raskin ([https://arxiv.org/abs/2603.26925 arXiv:2603.26925]) formulated conjectures on the canonical filtration and decomposition of automorphic functions by nilpotent orbits of the dual group, including a rationality conjecture and their Conjectures 3.4.5 and 3.4.6.

== The original results ==
* '''Restricted geometric Langlands in positive characteristic (24 September 2026).''' The manuscript states the ℓ-adic restricted geometric Langlands equivalence for smooth projective connected curves and connected reductive groups in characteristic ''p'' > 0 in two regimes: over an algebraic closure of a finite field under four stated Lie-theoretic conditions on the characteristic, and over arbitrary algebraically closed fields under those conditions together with ''p'' very good for the group and not dividing the order of the Weyl group. It describes this as proving the full-support conjecture of Gaitsgory and Raskin in these regimes.
* '''Arthur filtration and Ramanujan–Arthur decompositions (24 September 2026).''' One manuscript states that the canonical Arthur filtration on finitely supported unramified automorphic functions is defined over the rational numbers for split connected semisimple groups, establishing the rationality conjecture of Gaitsgory, Lafforgue and Raskin in that setting. A second states rational and ℓ-adic decompositions of cuspidal automorphic functions at every full finite level into summands indexed by nilpotent orbits of the dual group, which at trivial level proves Conjectures 3.4.5 and 3.4.6 of Gaitsgory, Lafforgue and Raskin. It also states that, for split connected adjoint absolutely simple groups, a cuspidal automorphic representation with one generic unramified local component is tempered at every unramified place.
* '''Ramanujan at ramified places for exceptional groups (5 October 2026).''' Theorem 1.1 of the manuscript states that for split connected adjoint absolutely simple groups of type ''G''<sub>2</sub>, ''F''<sub>4</sub>, ''E''<sub>6</sub>, ''E''<sub>7</sub> or ''E''<sub>8</sub> over a global function field, every local component of a globally generic cuspidal automorphic representation is tempered, with no restriction on the characteristic, the ramification depth or the local representation type.
* '''Arthur enhancements (5 October 2026).''' Assuming the finite-level Ramanujan–Arthur decomposition, a further manuscript constructs a global Arthur enhancement of every occurring cuspidal excursion parameter at a full finite level for split connected semisimple groups over global function fields, recovering the parameter by diagonal specialization on the whole Weil group. It states that it does not assert ellipticity, an Arthur-packet classification or a multiplicity formula.
* '''Tame Hecke eigensheaves (5 October 2026).''' Three manuscripts construct nonzero perverse Hecke eigensheaves with Borel level on curves of genus at least two: at one marked point for simple simply connected groups under four conditions on the characteristic, with parameter a Zariski-dense local system with tame regular-unipotent monodromy; at two or more marked points for SL<sub>''n''</sub> with ''p'' > ''n''; and with a Frobenius-compatible Weil structure for SL<sub>''n''</sub> with ''p'' > ''n'' at one marked point.
** ''Method, as described:'' the full-support statement is proved by showing that geometric specialization from characteristic zero cannot annihilate the nonzero spectral summand attached to a component, using Gaitsgory and Raskin's characteristic-zero comparison. The ramified Ramanujan theorem extends the unramified theorem of the companion manuscript, comparing monodromy operators through the Genestier–Lafforgue parameters and the essentially tempered completions of Gan, Harris and Sawin.
* '''Sources.''' Commit <code>1581d4c</code> of the openai/math repository, folders [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic-September-24-2026 The restricted geometric Langlands equivalence in positive characteristic], [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Rationality-of-the-Canonical-Unramified-Arthur-Filtration-September-24-2026 Rationality of the canonical unramified Arthur filtration], [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level-September-24-2026 Ramanujan–Arthur decompositions of cuspidal functions at full finite level], [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Temperedness-at-ramified-places-for-globally-generic-exceptional-groups-October-5-2026 Temperedness at ramified places for globally generic exceptional groups], [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Global-Arthur-Enhancements-of-Cuspidal-Excursion-Parameters-October-5-2026 Global Arthur enhancements of cuspidal excursion parameters], [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Constructible-tame-Hecke-eigensheaves-in-positive-characteristic-October-5-2026 Constructible tame Hecke eigensheaves in positive characteristic], [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Tame-Hecke-Eigensheaves-with-Several-Marked-Points-October-5-2026 Tame Hecke eigensheaves with several marked points], [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Frobenius-Structures-on-Tame-Hecke-Eigensheaves-October-5-2026 Frobenius structures on tame Hecke eigensheaves].
* '''Formalization.''' The collection has no Lean scope note in <code>lean/docs/</code> for this family.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 18 October 2026)
|-
| 2026-10-16 || J. Jardine, Lattice24, referee reviews [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_017.json rank_017] (restricted equivalence), [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_269.json rank_269] (Arthur filtration), [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_083.json rank_083] (Ramanujan–Arthur decompositions), [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_250.json rank_250] (ramified temperedness), [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_098.json rank_098] (Arthur enhancements), [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_247.json rank_247] (constructible eigensheaves), [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_315.json rank_315] (several marked points), [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_431.json rank_431] (Frobenius structures) in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995]) || Review (AI-assisted audit) || All eight reviews report no gap. Their notes record reliance on cited results that the reviews could not check offline, among them Gaitsgory and Raskin's characteristic-zero equivalence, a central-sheaf trace identity of Dhillon and Taylor, a corollary of Ciubotaru and Harris, the essentially tempered completions of Gan, Harris and Sawin, and a functional equation from Lomelí's framework. The review of the Arthur-enhancement manuscript notes that it is only as strong as the companion decomposition theorem, and the review of the restricted equivalence notes a harmless omitted remark on the valuative criterion for algebraic spaces. || Part of a completed audit of the collection's manuscripts without Lean; the reviews are published under CC BY 4.0.
|}

== Notes ==
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]