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Math-2799 · mathematics · Number theory

Rationality of the canonical unramified Arthur filtration

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let X/Fq be a smooth projective geometrically connected curve and G/Fq split connected semisimple. Assume that the Lie algebra admits an invariant nondegenerate symmetric form nondegenerate on every Levi center, Chevalley restriction is an isomorphism for every Levi, semisimple Lie-element scheme centralizers are Levi subgroups, and every nilpotent Lie element over every extension belongs to the Lie algebra of the unipotent radical of an extension-defined parabolic. Let $A_E$ be finitely supported E-valued functions on the finite-field G-bundle isomorphism classes, $E=Qbar_\ell $. For any closed invariant subset Y of the dual Lie-algebra nilpotent cone, let $F_Y$ $A_E$ be the image of categorical Frobenius trace from the corresponding derived-Satake nilpotent-support subcategory. Must E $tensor_Q$ $(A_Q$ intersect $F_Y$ $A_E)$ map isomorphically to $F_Y$ $A_E$ for every ell different from p, yielding a common rational filtration independent of ell?

Short English statement

Let X/Fq be a smooth projective geometrically connected curve and G/Fq split connected semisimple. Assume that the Lie algebra admits an invariant nondegenerate symmetric form nondegenerate on every Levi center, Chevalley restriction is an isomorphism for every Levi, semisimple Lie-element scheme centralizers are Levi subgroups, and every nilpotent Lie element over every extension belongs to the Lie algebra of the unipotent radical of an extension-defined parabolic. Let $A_E$ be finitely supported E-valued functions on the finite-field G-bundle isomorphism classes, $E=Qbar_\ell $. For any closed invariant subset Y of the dual Lie-algebra nilpotent cone, let $F_Y$ $A_E$ be the image of categorical Frobenius trace from the corresponding derived-Satake nilpotent-support subcategory. Must E $tensor_Q$ $(A_Q$ intersect $F_Y$ $A_E)$ map isomorphically to $F_Y$ $A_E$ for every ell different from p, yielding a common rational filtration independent of ell?

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Math-2799-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2799

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 014; 2026-09-24).

Rationality of the canonical unramified Arthur filtration

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 014; 2026-09-24), the asserted result is that the canonical unramified Arthur filtration on all finitely supported automorphic functions is rational under the restricted theory's characteristic hypotheses for split connected semisimple groups.

Mathematical statement of the claimed result

Manuscript 1.
Let $X/\mathbb F_q$ be smooth projective geometrically connected and $G$ split connected semisimple. On $H=G_{\overline{\mathbb F}_q}$ assume an invariant nondegenerate Lie-algebra form nondegenerate on all Levi centers, Chevalley restriction for every Levi, semisimple Lie-algebra centralizers being Levi, and every nilpotent element over any extension lying in a parabolic unipotent-radical Lie algebra. For $E=\overline{\mathbb Q}_{\ell}$, $\ell\ne\operatorname{char}\mathbb F_q$, let $S$ be bundle isomorphism classes over $\mathbb F_q$ and $\mathcal A_E=E^{(S)}$. For a closed invariant $Y$ in the pinned dual-group nilpotent cone, use the derived-Satake support category $\mathcal C_Y$ and Frobenius $\Phi$ to define \[ \mathcal F_Y\mathcal A_E=\operatorname{im}(H^0\operatorname{Tr}(\Phi,\mathcal C_Y)\to\mathcal A_E). \] The claim says this entire canonical filtration is rational: \[ E\otimes_{\mathbb Q}(\mathcal A_{\mathbb Q}\cap\mathcal F_Y\mathcal A_E) \xrightarrow{\sim}\mathcal F_Y\mathcal A_E \] for every such $Y$. The intersection is inside $\mathcal A_E$. Noncuspidal functions are included, and supported traces are defined by their image without assuming concentration in degree zero.

Outline of the argument

For manuscript 1 (family 014; 2026-09-24), a bilinear detector tests supported spectral local cohomology against finite evaluations. Orbit distributions, compact Weil lifts, and noncancellation separate the nilpotent supports, and rational descent then includes the noncuspidal functions.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 014; 2026-09-24).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

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