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Rationality of the Canonical Unramified Arthur Filtration

OpenAI · September 24, 2026

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Abstract

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.

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work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Digests and expositions

  • Platzi, Rationality of the canonical Arthur filtration (OpenAI Math Preprints, lesson 30, undated). The publicly accessible lesson summary outlines the paper’s bilinear measure criterion, invariance under coefficient automorphisms and rational descent for the filtration, including noncuspidal functions.

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Rationality-of-the-Canonical-Unramified-Arthur-Filtration-September-24-2026,
  author = {{OpenAI}},
  title = {{Rationality of the Canonical Unramified Arthur Filtration}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Rationality-of-the-Canonical-Unramified-Arthur-Filtration-September-24-2026/paper.pdf}{OAI:Rationality-of-the-Canonical-Unramified-Arthur-Filtration-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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