Short··0:20·Number theory
Restricted Geometric Langlands: OpenAI Claims Characteristic p
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About this video
The geometric Langlands program predicts a deep match between automorphic objects on a curve and Galois-side data; in characteristic p its "restricted" form was conjectured by Gaitsgory and Raskin. OpenAI claims a proof of that restricted equivalence for connected reductive groups on smooth projective curves (under stated conditions on p), plus the generalized Ramanujan conjecture for globally generic cuspidal representations of split adjoint exceptional groups over function fields, and global Arthur enhancements assuming a finite-level Ramanujan–Arthur decomposition. These are OpenAI's claims and have not yet been peer reviewed.
What you see is a real computed example of what "Ramanujan" (tempered) means, for the simplest such group, PGL2, where it was already known (Drinfeld). The object is the automorphic form of the elliptic curve y² = x³ + tx + 1 over the function field F13(t). Each ring is one of its 91 places of degree 1 or 2. The two gold points on a ring are that place's Satake parameters, computed from exact point counts, and every one of them sits on the unit circle. The pink points belong to the constant function, which is automorphic but not cuspidal. Its parameters q^±½ sit off the circle, and its Arthur SL2 accounts for exactly that shift. OpenAI's Ramanujan claim is for exceptional groups such as G2 and E8, which the picture does not show.
The animation illustrates the result; it is not a proof.
Papers:
- The Restricted Geometric Langlands Equivalence in Positive Characteristic: github.com/openai/math/blob/main/preprints/The-Restricted-Geo…
- Temperedness at ramified places for globally generic exceptional groups: github.com/openai/math/blob/main/preprints/Temperedness-at-ra…
- Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level: github.com/openai/math/blob/main/preprints/Ramanujan-Arthur-D…
- Global Arthur Enhancements of Cuspidal Excursion Parameters: github.com/openai/math/blob/main/preprints/Global-Arthur-Enha…
- Rationality of the Canonical Unramified Arthur Filtration: github.com/openai/math/blob/main/preprints/Rationality-of-the…
- Constructible tame Hecke eigensheaves in positive characteristic: github.com/openai/math/blob/main/preprints/Constructible-tame…
- Tame Hecke Eigensheaves with Several Marked Points: github.com/openai/math/blob/main/preprints/Tame-Hecke-Eigensh…
- Frobenius Structures on Tame Hecke Eigensheaves: github.com/openai/math/blob/main/preprints/Frobenius-Structur…
Original visuals (three.js) and synthesized sound: every place plucks a note pitched by its Satake angle.
Gradient Damage: mathematics you can see, equations you can hear.
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Every result explained: gradientdamage.com/p/014
About the problem
Number theory·OpenAI Math·Result 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 𝔽̅q 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∤ |WG|. 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.
Topics
geometric LanglandsLanglands programrestricted geometric LanglandsRamanujan conjecturegeneralized Ramanujan conjectureSatake parameterstempered representationsArthur parametersautomorphic formsfunction fieldscharacteristic pGaitsgoryRaskinLafforgueDrinfeldexceptional groupselliptic curveHecke eigensheavesOpenAI mathAI proofAI mathematicsmath visualizationmaths shortsGradient Damage
