The Gaitsgory-Lafforgue-Raskin Ramanujan-Arthur decomposition of cuspidal automorphic functions over function fields

Let $F$ be the function field of a smooth projective geometrically connected curve over $\mathbb F_q$ , $G$ a split connected semisimple group, and $C_{\mathbb Q}$ the space of rational-valued cuspidal functions on $G(F)\backslash G(\mathbb A_F)/G(\mathcal O_{\mathbb A})$ . For a nilpotent orbit $\mathcal O$ of the dual group with Jacobson-Morozov map $\phi_{\mathcal O}$ , let $C_{\mathcal O}$ be the functions whose Hecke eigenvalues at every place $x$ have Satake class in $q_x^{1/2}$ -twisted $\phi_{\mathcal O}$ times a compact element of the centralizer, the same orbit at every place. Gaitsgory, Lafforgue and Raskin conjectured (Conjectures 3.4.5 and 3.4.6, the rational and the $\ell$ -adic Ramanujan-Arthur conjecture) that the cuspidal space decomposes accordingly. Is $\bigoplus_{\mathcal O}C_{\mathcal O}=C$ ?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Automorphic forms over function fields; Arthur parameters
Posed by: Dennis Gaitsgory, Vincent Lafforgue and Sam Raskin
Year posed: 2026
Years open: 0y
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 38 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for every $X,G,\ell$ and every effective divisor $D$ , the rational and $\overline{\mathbb Q}_\ell$ cuspidal spaces at full level $D$ are the direct sums of their orbit-indexed summands; $D=0$ gives GLR Conjectures 3.4.5-3.4.6. Corollary 1.2: for split adjoint absolutely simple $G$ , a cuspidal representation generic and unramified at one place is tempered at every unramified place. The October 5 companion constructs, from this, a commuting algebraic $SL_2$ enhancing each occurring cuspidal excursion parameter at full level; it does not assert ellipticity, Arthur packets or multiplicities, so V. Lafforgue's Arthur conjecture is not settled.

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. This entry rests on the September 24 Ramanujan-Arthur decomposition paper; the October 5 global Arthur enhancement paper builds on it.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 were read against GLR Conjectures 3.4.5-3.4.6. The theorem is stated for every curve, every split connected semisimple $G$ , every full level $D$ and every $\ell$ , with no characteristic hypothesis; at $D=0$ it is the conjecture. Inputs are V. Lafforgue's excursion parameters, L. Lafforgue's purity theorem for $GL_n$ , and the Ciubotaru-Harris criterion for the corollary. The proof was not refereed. No Lean formalization exists for this family.

The Gaitsgory-Lafforgue-Raskin Ramanujan-Arthur decomposition of cuspidal automorphic functions over function fields

What was actually shown

What the AI did

Verification

Sources

Discussion