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Math-2782 · mathematics · Number theory

Global Arthur enhancements of occurring excursion parameters

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Assume the orbit-indexed Ramanujan-Arthur decomposition of cuspidal functions at full finite level. For a split connected semisimple group over a finite field and a cuspidal excursion parameter sigma on the Weil group of the unramified open curve, does there exist a dual-group SL2 homomorphism phi and a continuous homomorphism tau to its full centralizer, both over one finite $Q_\ell $ extension, such that tau has reductive Zariski closure and all Frobenius eigenvalues have modulus one under every complex embedding, while after a single global conjugation $\sigma (w)=\tau (w)$ $\phi (\operatorname{diag}(a^deg(w),a^-deg(w)))$ for every Weil element w, including inertia, with a matching choice of square root of q?

Short English statement

Assume the orbit-indexed Ramanujan-Arthur decomposition of cuspidal functions at full finite level. For a split connected semisimple group over a finite field and a cuspidal excursion parameter sigma on the Weil group of the unramified open curve, does there exist a dual-group SL2 homomorphism phi and a continuous homomorphism tau to its full centralizer, both over one finite $Q_\ell $ extension, such that tau has reductive Zariski closure and all Frobenius eigenvalues have modulus one under every complex embedding, while after a single global conjugation $\sigma (w)=\tau (w)$ $\phi (\operatorname{diag}(a^deg(w),a^-deg(w)))$ for every Weil element w, including inertia, with a matching choice of square root of q?

Sources

Statement licence: CC-BY-4.0 · Source file

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Math-2782-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2782

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; 2026-10-05).

Global Arthur enhancements of occurring excursion parameters

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; 2026-10-05), the asserted result is that assuming the stated finite-level decomposition, each occurring cuspidal excursion parameter for a split semisimple function-field group has a global Arthur enhancement with one algebraic $\mathrm{SL}_2$ and a commuting Weil action, including inertia.

Mathematical statement of the claimed result

Manuscript 1.
Let $X/\mathbb F_q$ be smooth projective geometrically connected, $G/\mathbb F_q$ split connected semisimple, $D$ an effective divisor, $U=X\setminus\operatorname{supp}D$ and $K_D=\ker(G(\mathcal O_{\mathbb A})\to G(\mathcal O_D))$. Fix $\ell\ne\operatorname{char}\mathbb F_q$, $k=\overline{\mathbb Q}_{\ell}$, an embedding $j:\overline{\mathbb Q}\to k$ and $a=j(\sqrt q)$. For an occurring character $\nu$ of the excursion algebra on cuspidal functions at full level $K_D$, write $\sigma_{\nu}:\pi_1(U)\to\widehat G(k)$ for its parameter and $\Gamma=W_U$ for the Weil subgroup.

The theorem assumes the companion finite-level decomposition: one nilpotent orbit $\mathcal O_{\nu}$ works at all good points, and each semisimple Frobenius class has the form $[\phi_{\mathcal O_{\nu}}(\operatorname{diag}(a^{d_x},a^{-d_x}))c_x]$, with $c_x$ commuting, algebraic and compact at every complex embedding. Under this assumption it claims a global algebraic $\phi_{\nu}:\operatorname{SL}_2\to\widehat G$ and continuous $\tau_{\nu}:\Gamma\to Z_{\widehat G}(\phi_{\nu}(\operatorname{SL}_2))(k)$, defined over a finite $\ell$-adic extension, with reductive Zariski closure, such that after one fixed conjugation \[ \sigma_{\nu}(w)=\tau_{\nu}(w)\phi_{\nu}(\operatorname{diag}(a^{\deg w},a^{-\deg w}))\quad(w\in\Gamma). \] The differential of $\phi_{\nu}$ sends the standard upper nilpotent matrix into $\mathcal O_{\nu}$. For every rational-form algebraic representation and closed $x\in U$, eigenvalues of $\tau_{\nu}(\operatorname{Frob}_x)$ are algebraic and have absolute value one at every complex embedding. The full centralizer may be disconnected. Ellipticity, packet classification and a multiplicity formula are not asserted.

Outline of the argument

For manuscript 1 (family 014; 2026-10-05), shtuka cohomology supplies a finite-type cyclic support carrying full excursion data. Frobenius-compatible coherent models of Iwahori categories at several places identify an open-orbit condition, which assembles a single commuting Weil and $\mathrm{SL}_2$ pair.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 014; 2026-10-05).

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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