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Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan

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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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What the papers state

Theorem 1.1 (Finite-level Ramanujan–Arthur decomposition [Ramanujan-Arthur Decompositions of Cuspidal…, Theorem 1.1])

In Global Arthur Enhancements of Cuspidal Excursion Parameters, 5 October 2026

The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.

For every occurring excursion character \nu, there is a unique nilpotent L-orbit \mathcal O_\nu\subset\Lie L such that, at every closed point x\in U, the semisimple class of \sigma_\nu(\Frob_x) has the form

\left[ \phi_{\mathcal O_\nu} \begin{pmatrix}a^{d_x}&0\\0&a^{-d_x}\end{pmatrix}c_x \right].

Here \phi_{\mathcal O_\nu}:\SL_2\to L is a Jacobson–Morozov homomorphism for \mathcal O_\nu and c_x is semisimple, centralizes its image, and is algebraic and conjugate into a compact subgroup at every complex embedding. The same orbit occurs at every x.

Corollary 1.2 (Unramified generalized Ramanujan)

In Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level, 24 September 2026

Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.

Let F be the function field of a smooth projective geometrically connected curve over a finite field, and let G/F be split connected adjoint absolutely simple. Let \pi=\bigotimes'_x\pi_x be a complex cuspidal automorphic representation of G(\mathbb A_F). If \pi_v is unramified and generic at one place v, then \pi_x is tempered at every place x at which it is unramified. In particular, every globally generic cuspidal automorphic representation of G(\mathbb A_F) is tempered at every unramified place.

The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.

Nine more possible matches

  • Theorem 1.1 in Temperedness at ramified places for globally generic exceptional groups, 5 October 2026.
  • Theorem 1.1 in Frobenius Structures on Tame Hecke Eigensheaves, 5 October 2026.
  • Theorem 1.2 (Global Arthur enhancement) in Global Arthur Enhancements of Cuspidal Excursion Parameters, 5 October 2026.
  • Theorem 1.1 in Tame Hecke Eigensheaves with Several Marked Points, 5 October 2026.
  • Theorem 1.1 in Constructible tame Hecke eigensheaves in positive characteristic, 5 October 2026.
  • Theorem 1.4 (Full support) in The Restricted Geometric Langlands Equivalence in Positive Characteristic, 24 September 2026.
  • Theorem 1.5 (Rational spectral support) in The Restricted Geometric Langlands Equivalence in Positive Characteristic, 24 September 2026.
  • Theorem 1.2 in Rationality of the Canonical Unramified Arthur Filtration, 24 September 2026.
  • Theorem 1.1 in Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level, 24 September 2026.
Reported proof verification
The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations
Eight current papers, each readable here as a page beside its PDF.

Papers

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Global Arthur Enhancements of Cuspidal Excursion Parameters

5 October 2026, current edition.

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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.

Rationality of the Canonical Unramified Arthur Filtration

24 September 2026, current edition.

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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.

Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level

24 September 2026, current edition.

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We prove rational and \(\overline{\mathbb Q}_\ell\) 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.

Temperedness at ramified places for globally generic exceptional groups

5 October 2026, current edition.

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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.

The Restricted Geometric Langlands Equivalence in Positive Characteristic

24 September 2026, current edition.

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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.

Constructible tame Hecke eigensheaves in positive characteristic

5 October 2026, current edition.

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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.

Tame Hecke Eigensheaves with Several Marked Points

5 October 2026, current edition.

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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.

Frobenius Structures on Tame Hecke Eigensheaves

5 October 2026, current edition.

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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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