Rationality and prime-independence of the canonical Arthur filtration
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Is the canonical unramified Arthur filtration on finitely supported automorphic functions defined over Q and independent of ℓ, including its noncuspidal part? The target assumes a split connected semisimple group over a finite field with the four specified characteristic conditions: a nondegenerate invariant Lie-algebra form, Chevalley restriction for every Levi, Levi centralizers of semisimple elements and suitable parabolics for nilpotent elements.
Why it matters and what counts as progress
Why it matters: Rational descent would make the full canonical Arthur filtration independent of the auxiliary ℓ-adic coefficient choice, including noncuspidal functions.
What counts as progress: Prove rational descent for the full filtration under these conditions, or remove one of the conditions with a valid descent argument.
How the problem developed
Dennis Gaitsgory, Vincent Lafforgue, and Sam Raskin, with David Kazhdan Formulated rationality of the canonical unramified Arthur filtration as Conjecture 2.7.3; their paper described it as a difficult open conjecture.
- 24 Sept 2026 · Reported claim [2]
OpenAI The preprint claims rational descent and prime-independence for the full finitely supported unramified automorphic-function space, including noncuspidal functions, under four stated characteristic hypotheses.
AI contribution
Rationality of the Canonical Unramified Arthur Filtration
Assuming the four stated characteristic conditions on the split connected semisimple group G over a finite field, every canonical unramified Arthur-filtration subspace on the full space of finitely supported automorphic functions is defined over Q and is independent of ℓ, including noncuspidal functions.
Sources
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Publication dates and editorial changes.
OpenAI reports rationality of the canonical Arthur filtration