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Math-2781 · mathematics · Number theory

Full finite-level Ramanujan-Arthur decomposition

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let X/Fq be smooth projective geometrically connected, G/Fq split connected semisimple, and D any effective divisor. Consider compactly supported cuspidal functions on $G(F)\backslash G(A_F)$ at full congruence level D. For each nilpotent orbit O of the dual group choose its SL2 homomorphism phi, and at a good degree-d place require the unramified Satake class to lie in $\phi (\operatorname{diag}(q^{d/2},q^{-d/2}))$ times a semisimple element of the centralizer whose eigenvalues in every algebraic representation have modulus one under every complex embedding. Must the associated orbit-indexed cuspidal subspaces give a direct-sum decomposition of the whole space, both over Q and over $Qbar_\ell $, with no restriction on D or characteristic?

Short English statement

Let X/Fq be smooth projective geometrically connected, G/Fq split connected semisimple, and D any effective divisor. Consider compactly supported cuspidal functions on $G(F)\backslash G(A_F)$ at full congruence level D. For each nilpotent orbit O of the dual group choose its SL2 homomorphism phi, and at a good degree-d place require the unramified Satake class to lie in $\phi (\operatorname{diag}(q^{d/2},q^{-d/2}))$ times a semisimple element of the centralizer whose eigenvalues in every algebraic representation have modulus one under every complex embedding. Must the associated orbit-indexed cuspidal subspaces give a direct-sum decomposition of the whole space, both over Q and over $Qbar_\ell $, with no restriction on D or characteristic?

Sources

Statement licence: CC-BY-4.0 · Source file

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Math-2781-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2781

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).

Full finite-level Ramanujan-Arthur decomposition

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 cuspidal functions for split semisimple groups over function fields decompose by dual nilpotent orbits at every full finite level; for adjoint absolutely simple groups, one generic unramified component forces temperedness at every unramified place.

Mathematical statement of the claimed result

Manuscript 1.
Let $F=\mathbb F_q(X)$ for a smooth projective geometrically connected curve, $G/\mathbb F_q$ split connected semisimple, and $D$ any effective divisor. Write $K_D=\ker(G(\mathcal O_{\mathbb A})\to G(\mathcal O_D))$ and $C_{D,\mathbb Q}$ for compactly supported rational cuspidal functions on $G(F)\backslash G(\mathbb A_F)/K_D$. For each dual-group nilpotent orbit $\mathcal O$, choose its Jacobson--Morozov $\phi_{\mathcal O}:\operatorname{SL}_2\to\widehat G$. Let $C_{D,\mathcal O,\mathbb Q}$ consist of functions whose spherical Hecke spectra, at every good point $x$, have semisimple class \[ [\phi_{\mathcal O}(\operatorname{diag}(q_x^{1/2},q_x^{-1/2}))c], \] where $c$ centralizes $\phi_{\mathcal O}$ and is algebraic and compact under every complex embedding. The same orbit is required at every good point. The claimed rational and $\ell$-adic decompositions are \[ C_{D,\mathbb Q}=\bigoplus_{\mathcal O} C_{D,\mathcal O,\mathbb Q},\qquad C_{D,\ell}=\bigoplus_{\mathcal O} C_{D,\mathcal O,\ell}, \] with $\ell\ne\operatorname{char}\mathbb F_q$ and scalar extensions from $\mathbb Q$. For split adjoint absolutely simple $G/F$, a cuspidal representation generic and unramified at one place is further claimed tempered at every unramified place. Ramified temperedness and a full multiplicity formula are not asserted here.

Outline of the argument

For manuscript 1 (family 014; 2026-09-24), iwahori translation eigencharacters are analyzed before taking the Weyl quotient. Frobenius traces of central kernels and Springer tests bound the extreme Bernstein exponent, while trace identities synchronize the governing nilpotent orbit across places and coefficient embeddings.

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.

Sources

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