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The generalized Ramanujan conjecture for globally generic cuspidal representations of exceptional groups over function fields

Let FF be the function field of a curve over Fq\mathbb F_q and GG a split connected reductive group. The generalized Ramanujan conjecture, in its generic form (within Arthur's framework; see Shahidi), predicts that every globally generic cuspidal automorphic representation π=⨂v′πv\pi=\bigotimes'_v\pi_v of G(AF)G(\mathbb A_F) is tempered at every place vv. Over function fields this was known for GL2GL_2 (Drinfeld), GLnGL_n (L. Lafforgue), and split classical and quasi-split unitary groups (Lomeli); for exceptional groups only conditional unramified results (Sawin-Templier, Ciubotaru-Harris) were available. For split adjoint groups of type G2,F4,E6,E7,E8G_2,F_4,E_6,E_7,E_8 over a global function field, is every globally generic cuspidal automorphic representation tempered at every place?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Automorphic representations over function fields; temperedness
Posed by
Generic form of the generalized Ramanujan conjecture (Arthur's framework; Shahidi)
Year posed
—
Years open
—
Solved
2026-10-05
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
34 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for GG split connected adjoint absolutely simple of type G2,F4,E6,E7G_2,F_4,E_6,E_7 or E8E_8 over a global function field, every complex globally generic cuspidal automorphic representation is tempered at every place. The companion (September 24) gives temperedness at unramified places for all split adjoint absolutely simple groups from genericity at one unramified place. Nothing is claimed over number fields, for non-generic representations, or for ramified places of classical groups beyond what Lomeli already proved.

What the AI did

The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The global input is the unramified Ramanujan corollary of the family's September 24 decomposition paper.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the generic Ramanujan conjecture. It covers split adjoint exceptional groups over function fields, every place including ramified ones, with no restriction on characteristic or ramification depth; this is a special case of the conjecture (one group class, function fields), hence Partial. The unramified places come from the companion decomposition paper; the ramified places use Genestier-Lafforgue local-global compatibility and Gan-Harris-Sawin. The proof was not refereed. No Lean formalization exists for this family.

Sources

Changelog1 change

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