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Math-2788 · mathematics · Number theory

Tame eigensheaves at several marked points

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let $n>=2$ and $p>n$. Let X be a smooth projective geometrically connected genus-at-least-two curve over an algebraic closure of a finite field, with at least two distinct finite-field rational marked points. For every Zariski-dense geometric PGLn parameter on the punctured curve over a finite extension of $Q_\ell $, killing wild inertia and having tame $\exp (t_\ell $ $N_x)$ monodromy for an arbitrary nilpotent $N_x$ at each mark, must there be a nonzero locally constructible perverse sheaf on SLn-bundles with Borel reduction at all marks, parabolic global-nilpotent singular support, and coherent all-leg Hecke eigenisomorphisms with collision fusion? Zero residues are allowed and no Frobenius descent is required.

Short English statement

Let $n>=2$ and $p>n$. Let X be a smooth projective geometrically connected genus-at-least-two curve over an algebraic closure of a finite field, with at least two distinct finite-field rational marked points. For every Zariski-dense geometric PGLn parameter on the punctured curve over a finite extension of $Q_\ell $, killing wild inertia and having tame $\exp (t_\ell $ $N_x)$ monodromy for an arbitrary nilpotent $N_x$ at each mark, must there be a nonzero locally constructible perverse sheaf on SLn-bundles with Borel reduction at all marks, parabolic global-nilpotent singular support, and coherent all-leg Hecke eigenisomorphisms with collision fusion? Zero residues are allowed and no Frobenius descent is required.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2788-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2788

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).

Tame eigensheaves at several marked points

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 for $\mathrm{SL}_n$ with $p>n$, dense tame parameters with arbitrary unipotent Jordan types at several marked points admit nonzero Borel-level perverse eigensheaves retaining tensor and fusion compatibility.

Mathematical statement of the claimed result

Manuscript 1.
Let $n\ge2$, $k=\overline{\mathbb F}_q$ of characteristic $p>n$, $\ell\ne p$, and $E=\overline{\mathbb Q}_{\ell}$. Take a smooth projective geometrically connected $X_0/\mathbb F_q$ of genus at least two with distinct rational $x_1,\ldots,x_s$, $s\ge2$, and set $X=(X_0)_k$, $D=\sum_ix_i$, $U=X\setminus|D|$. A continuous Zariski-dense $\rho:\pi_1^{\rm et}(U)\to\operatorname{PGL}_n(L)$, $L/\mathbb Q_{\ell}$ finite, is assumed trivial on wild inertia and has \[ \rho(\gamma)=\exp(t_{\ell,i}(\gamma)N_i)\quad(\gamma\in I_{x_i}), \] with any nilpotent $N_i$, including zero. On $\mathcal A=\operatorname{Bun}_{\operatorname{SL}_n,B,D}(X)$ the claim supplies a nonzero locally constructible perverse geometric sheaf $M$ with singular support in the generically nilpotent Higgs cone with Borel-unipotent residues, and \[ \operatorname{Hecke}_{I,(V_i)}(M)\simeq M\boxtimes\mathop{\boxtimes}_{i\in I}(V_i)_{\rho} \] for all finite leg sets and dual-group representations. Eigenmaps are coherent for units, convolution, permutations and every collision diagonal, with relative Satake normalization $E[d_{\lambda}](d_{\lambda}/2)$ and no extra moving-leg shift.

Outline of the argument

For manuscript 1 (family 014; 2026-10-05), a curve with several separating nodes realizes all marked types simultaneously. Nilpotent cotangent dimension bounds and transverse modifications control specialization, after which commuting unipotent torus actions remove all flag enhancements without destroying the object.

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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