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Browse › Number theory › 014 Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan

Temperedness at ramified places for globally generic exceptional groups

OpenAI · October 5, 2026

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Abstract

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.

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Cite

@misc{OAI:Temperedness-at-ramified-places-for-globally-generic-exceptional-groups-October-5-2026,
  author = {{OpenAI}},
  title = {{Temperedness at ramified places for globally generic exceptional groups}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Temperedness-at-ramified-places-for-globally-generic-exceptional-groups-October-5-2026/ramified-ramanujan.pdf}{OAI:Temperedness-at-ramified-places-for-globally-generic-exceptional-groups-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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