Math-2778 · mathematics · Number theory

Generic Ramanujan for exceptional groups over function fields

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.

Over every global function field F, is every local component of every complex globally generic cuspidal representation of a split connected adjoint absolutely simple group of type G2,F4,E6,E7 or E8 tempered, including arbitrarily ramified places and every characteristic?

Short English statement

Sources

S1 Original question source or mathematical context. https://www.claymath.org/library/monographs/cmim04c.pdf.
S2 Original question source or mathematical context. https://doi.org/10.4171/owr/2005/39.
S3 Original question source or mathematical context. https://doi.org/10.4171/owr/2024/54.
S4 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Temperedness-at-ramified-places-for-globally-generic-exceptional-groups-October-5-2026/README.md.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2778-A1 · Literature report · LLM-generated · unknown · 2026-10-06

OpenAI claim summary for Math-2778

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 014; 2026-10-05).

Generic Ramanujan for exceptional groups over function fields

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 014; 2026-10-05), the asserted result is that globally generic cuspidal representations of split adjoint exceptional groups over function fields are tempered at ramified as well as unramified places, in every characteristic.

Mathematical statement of the claimed result

Manuscript 1.

Let $F$ be a global function field and $G/F$ split connected adjoint absolutely simple of one of the exceptional types $G_2,F_4,E_6,E_7,E_8$. For every complex globally generic cuspidal automorphic representation $\pi=\bigotimes'_v\pi_v$ of $G(\mathbb A_F)$, the claimed conclusion is \[ \pi_v\text{ is tempered for every place }v. \] Ramified places are included. The statement imposes no characteristic, ramification-depth or local-representation-type restriction.

Outline of the argument

For manuscript 1 (family 014; 2026-10-05), the companion unramified result supplies global purity. Local functional equations compare semisimple parameters independently of monodromy; regularity of the adjoint local factor then identifies the monodromy orbit needed to apply the local temperedness criterion.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 014; 2026-10-05).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

OpenAI, Temperedness at ramified places for globally generic exceptional groups, 2026-10-05; citation key OAI:Temperedness-at-ramified-places-for-globally-generic-exceptional-groups-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
OpenAI catalogue entry.

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Record details

Subjects: Number theory
Collection: Top Open Problems
Mathematics rank: 2778