Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan

What OpenAI claims

Proves the restricted geometric Langlands equivalence for connected reductive groups on smooth projective connected curves over 𝔽̅_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∤ |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.

Papers (8)

1. Global Arthur Enhancements of Cuspidal Excursion Parameters

   5 October 2026 · Preprint, not peer reviewed · PDF

   Under the finite-level Ramanujan–Arthur decomposition stated in Theorem 1.1, we construct a global Arthur enhancement for every occurring cuspidal excursion parameter at a specified full finite level for a split connected semisimple group over a global function field. A single algebraic SL2 and a commuting Weil centralizer map recover the given parameter by diagonal specialization on the entire Weil group, including inertia. The result does not assert ellipticity, Arthur-packet classification, or a multiplicity formula.

2. Rationality of the Canonical Unramified Arthur Filtration

   24 September 2026 · Preprint, not peer reviewed · PDF

   Under the characteristic hypotheses of restricted geometric Langlands theory, we prove that the canonical Arthur filtration on finitely supported unramified automorphic functions is defined over ℚ for split connected semisimple groups. The result includes the noncuspidal part and every closed invariant nilpotent support, establishing the rationality conjecture of Gaitsgory–Lafforgue–Raskin in this setting.

3. Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level

   24 September 2026 · Preprint, not peer reviewed · PDF

   We prove rational and ℚ̅_ℓ decompositions of cuspidal automorphic functions for split semisimple groups over global function fields into subspaces indexed by nilpotent orbits of the dual group. The decompositions hold at every full finite level, with arbitrary divisor multiplicities. Each summand is governed by the same orbit at every unramified place and every complex embedding. At trivial level this proves Conjectures 3.4.5 and 3.4.6 of Gaitsgory–Lafforgue–Raskin. For split connected adjoint absolutely simple groups, we also prove that a cuspidal automorphic representation with one generic unramified local component is tempered at every unramified place. Thus globally generic cuspidal representations in this scope satisfy the unramified part of the generalized Ramanujan conjecture.

4. Temperedness at ramified places for globally generic exceptional groups

   5 October 2026 · Preprint, not peer reviewed · PDF

   We prove the generalized Ramanujan conjecture for globally generic cuspidal automorphic representations of split connected adjoint exceptional groups over global function fields. Every local component is tempered, without restrictions on characteristic or ramification depth. The proof extends the unramified Ramanujan theorem of a companion paper to all ramified places.

5. The Restricted Geometric Langlands Equivalence in Positive Characteristic

   24 September 2026 · Preprint, not peer reviewed · PDF

   We prove the ℚ̅_ℓ-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 𝔽̅_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] in these regimes.

6. Constructible tame Hecke eigensheaves in positive characteristic

   5 October 2026 · Preprint, not peer reviewed · PDF

   We construct nonzero locally constructible perverse Hecke eigensheaves with Borel level at one marked point on a smooth projective curve of genus at least two over an algebraic closure of a finite field. The parameter is a Zariski-dense geometric ℓ-adic local system with tame regular-unipotent monodromy. The group is simple and simply connected and satisfies four explicit Lie-theoretic characteristic hypotheses. The eigensheaves have parabolic nilpotent singular support, and their eigenisomorphisms are compatible with tensor products, permutations, and fusion.

7. Tame Hecke Eigensheaves with Several Marked Points

   5 October 2026 · Preprint, not peer reviewed · PDF

   For SLn in characteristic p > n, we construct nonzero locally constructible perverse Hecke eigensheaves with Borel level at two or more marked points on a smooth projective curve of genus at least two over an algebraic closure of a finite field. The parameter is a Zariski-dense geometric ℓ-adic PGLn-local system with unipotent tame monodromy; its nilpotent logarithms may have any Jordan type, including zero. The eigensheaves have parabolic nilpotent singular support, and their eigenisomorphisms retain the full tensor and fusion structure.

8. Frobenius Structures on Tame Hecke Eigensheaves

   5 October 2026 · Preprint, not peer reviewed · PDF

   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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.