The Gaitsgory-Raskin full-support conjecture for restricted geometric Langlands in positive characteristic

Let $X$ be a smooth projective connected curve over an algebraically closed field $k$ of characteristic $p>0$ , $G$ a connected reductive group, $\ell\ne p$ . Gaitsgory and Raskin constructed, under the characteristic hypotheses of the restricted theory of Arinkin-Gaitsgory-Kazhdan-Raskin-Rozenblyum-Varshavsky, a $\mathrm{QCoh}(Y)$ -linear equivalence from $\mathrm{Shv}_{\mathrm{Nilp}}(\mathrm{Bun}_G(X))$ onto $\mathrm{IndCoh}_{\mathrm{Nilp}}(Y')$ for an open-and-closed union $Y'$ of connected components of the restricted stack $Y=\mathrm{LS}^{\mathrm{restr}}_{\check G}(X)$ , and proved $Y'=Y$ for $GL_n$ . Their Conjecture 1.3.10 asserts that no component is missing. Is $Y'=Y$ , so that the restricted Langlands functor is an equivalence onto all of $\mathrm{IndCoh}_{\mathrm{Nilp}}(Y)$ ?

Result: Proved(see note)
Status: Partial result
AI contribution: AI-discovered
Method: Argument
Field: Geometric Langlands program; l-adic sheaves in positive characteristic
Posed by: Dennis Gaitsgory and Sam Raskin
Year posed: 2025
Years open: 1y
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: 30 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem (full support): under either regime, $Y'=Y$ , so $\mathbb L^{\mathrm{restr}}_G:\mathrm{Shv}_{\mathrm{Nilp}}(\mathrm{Bun}_G(X))\to\mathrm{IndCoh}_{\mathrm{Nilp}}(Y)$ is a $\mathrm{QCoh}(Y)$ -linear equivalence; a rational-coefficient form is also proved over $\overline{\mathbb F}_q$ . Consequences: nonzero localized categories at every spectral component, the full arithmetic trace formula in the finite-field regime, and nonzero Hecke eigenobjects for every parameter. It does not remove the characteristic hypotheses, and does not assert bounded constructibility or perversity of eigenobjects (the three October 5 eigensheaf companions construct perverse tame eigensheaves in specific settings).

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 family has eight manuscripts (September 24 to October 5); this entry rests on the September 24 restricted-equivalence paper.

Verification

No independent mathematician has checked this yet. Checked here: The full-support theorem was read against Gaitsgory-Raskin's conjecture. It is proved in two regimes: over $\overline{\mathbb F}_q$ under the four characteristic conditions referenced by Gaitsgory-Raskin (with a finite-field descent of $X$ and $G$ ), and over any algebraically closed field of characteristic $p$ assuming additionally that $p$ is very good and does not divide $|W_G|$ . The primed equivalence and the characteristic-zero input are cited from Gaitsgory-Raskin. Because the second regime carries extra hypotheses, Partial. The proof was not refereed. No Lean formalization exists for this family.

The Gaitsgory-Raskin full-support conjecture for restricted geometric Langlands in positive characteristic

What was actually shown

What the AI did

Verification

Sources

Discussion