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Math-2800 · mathematics · Number theory

Constructible tame eigensheaves with regular monodromy

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let X be a smooth projective geometrically connected curve of genus at least two over an algebraic closure of a finite field of characteristic p, with a finite-field rational marked point x, and let G be split simple simply connected. 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. For every Zariski-dense geometric dual-group parameter on X minus x over a finite extension of $Q_\ell $, ell different from p, killing wild inertia and having tame monodromy $\exp (t_\ell $ N) for regular nilpotent N, is there a nonzero locally constructible perverse ell-adic sheaf on G-bundles with Borel reduction at x, with parabolic nilpotent singular support and coherent all-leg Hecke eigenisomorphisms, including fusion at collisions? No arithmetic descent of the parameter is assumed.

Short English statement

Let X be a smooth projective geometrically connected curve of genus at least two over an algebraic closure of a finite field of characteristic p, with a finite-field rational marked point x, and let G be split simple simply connected. 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. For every Zariski-dense geometric dual-group parameter on X minus x over a finite extension of $Q_\ell $, ell different from p, killing wild inertia and having tame monodromy $\exp (t_\ell $ N) for regular nilpotent N, is there a nonzero locally constructible perverse ell-adic sheaf on G-bundles with Borel reduction at x, with parabolic nilpotent singular support and coherent all-leg Hecke eigenisomorphisms, including fusion at collisions? No arithmetic descent of the parameter is assumed.

Sources

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Math-2800-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2800

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).

Constructible tame eigensheaves with regular monodromy

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 tame regular-unipotent, geometrically dense parameters admit nonzero constructible perverse Hecke eigensheaves with one Borel-level point for the specified simply connected groups and characteristic hypotheses.

Mathematical statement of the claimed result

Manuscript 1.
Let $k=\overline{\mathbb F}_q$ have characteristic $p$, $E=\overline{\mathbb Q}_{\ell}$ with $\ell\ne p$, and let $X_0/\mathbb F_q$ be smooth projective geometrically connected of genus at least two with rational point $x_0$. Set $X=(X_0)_k$, $x=(x_0)_k$ and $U=X\setminus\{x\}$. Let $G_0$ be split simple simply connected, with Borel $B_0$, and write $G,B$ for base changes. The characteristic hypotheses are: a nondegenerate invariant symmetric form on $\mathfrak g$ whose restriction to the central Lie algebra of each Levi is nondegenerate; the Chevalley-restriction isomorphism $k[\mathfrak m]^M\simeq k[\operatorname{Lie}(T_M)]^{W_M}$ for every Levi; semisimple Lie-algebra centralizers are Levi subgroups; and over every extension field each nilpotent Lie-algebra element lies in a parabolic unipotent-radical Lie algebra.

For a continuous Zariski-dense parameter $\rho:\pi_1^{\rm et}(U)\to\widehat G(L)$, $L/\mathbb Q_{\ell}$ finite, require trivial wild inertia and tame monodromy \[ \rho(\gamma)=\exp(t_{\ell}(\gamma)N),\qquad \dim Z_{\widehat G}(N)=\operatorname{rank}(\widehat G), \] with nilpotent $N$ and $t_{\ell}:I_x\to\mathbb Z_{\ell}$. On the stack $\mathcal A=\operatorname{Bun}_{G,B,x}(X)$, the claim gives a nonzero locally constructible perverse $E$-adic sheaf $M$ with nilpotent parabolic singular support and \[ \operatorname{Hecke}_{I,(V_i)}(M)\simeq M\boxtimes\mathop{\boxtimes}_{i\in I}(V_i)_{\rho} \quad\hbox{on }\mathcal A\times U^I \] for every finite set $I$ and representations $V_i$ of $\widehat G$. The eigenisomorphisms respect units, convolution, permutations and collision-diagonal fusion. No Frobenius descent of $\rho$ is assumed. The nilpotent cone consists of Higgs fields with nilpotent generic value and residue in the Borel unipotent radical.

Outline of the argument

For manuscript 1 (family 014; 2026-10-05), the parameter is glued through a separating node to an unramified family. Two specializations and transverse Hecke cuts detect a nonzero flag-level object; unipotent monodromy on the enhancement torus permits descent to ordinary Borel level.

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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