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Global Arthur Enhancements of Cuspidal Excursion Parameters

OpenAI · October 5, 2026

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Abstract

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.

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Cite

@misc{OAI:Global-Arthur-Enhancements-of-Cuspidal-Excursion-Parameters-October-5-2026,
  author = {{OpenAI}},
  title = {{Global Arthur Enhancements of Cuspidal Excursion Parameters}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Global-Arthur-Enhancements-of-Cuspidal-Excursion-Parameters-October-5-2026/manuscript.pdf}{OAI:Global-Arthur-Enhancements-of-Cuspidal-Excursion-Parameters-October-5-2026}},
  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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