Rationality of the Gaitsgory-Lafforgue-Raskin canonical Arthur filtration
For a split connected semisimple group over and a curve , Gaitsgory, Lafforgue and Raskin define a canonical filtration, indexed by nilpotent orbits of the dual Lie algebra, on the space of finitely supported unramified automorphic functions with coefficients, using singular support of -adic sheaves and the trace of Frobenius. The space of functions has a natural rational form. Their Conjecture 2.7.3, formulated jointly with Kazhdan, asserts that the canonical filtration is defined over . Is the canonical Arthur filtration rational?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Automorphic functions over function fields; geometric Langlands
- Posed by
- Dennis Gaitsgory, Vincent Lafforgue and Sam Raskin, jointly with David Kazhdan
- 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
- 22 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Under the stated characteristic assumptions, the canonical Arthur filtration on finitely supported unramified automorphic functions is defined over , for split connected semisimple , including the noncuspidal part and every closed invariant nilpotent support. Using the companion decomposition at trivial level, each cuspidal filtration step is identified with the scalar extension of the corresponding rational summands (the common-scope form of GLR Conjectures 3.4.9 and 3.4.11). It does not treat small characteristic or nonsplit groups.
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.
Verification
No independent mathematician has checked this yet. Checked here: the main theorem was read against GLR Conjecture 2.7.3. It is proved for split connected semisimple groups under four explicit characteristic hypotheses (the restricted-theory assumptions), for the whole space including the noncuspidal part; outside those hypotheses the conjecture is not addressed, hence Partial. It uses the restricted geometric Langlands results and trace-of-Frobenius theorems as inputs. The proof was not refereed. No Lean formalization exists for this family.