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Math-2776 · mathematics · Number theory

Integral-point density on curve character varieties

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Let $X$ be a smooth connected complex algebraic curve and let $r\ge1$. Write $M_r(X)$ for the character variety of semisimple representations $\pi_1(X)\to\mathrm{SL}_r(\mathbb C)$, with its integral model obtained from a finite presentation of $\pi_1(X)$ and conjugation invariants. Representations have determinant one. If $\overline X\setminus X=\{x_1,\ldots,x_s\}$, choose meridians and quasi-unipotent conjugacy classes $C_i\subset\mathrm{SL}_r(\mathbb C)$; all their eigenvalues are roots of unity. Let $Y\subset M_r(X)_{\mathbb C}$ be the reduced locus where those meridians have the specified conjugacy classes, allowing nonsemisimple boundary matrices.

For a number field $K\subset\mathbb C$ containing the boundary eigenvalues, does there exist one finite extension $L/K$ for which \[ M_r(X)(\mathcal O_L)\cap Y(\mathbb C) \] is Zariski dense in $Y$, and hence in every irreducible component? Integrality means extension to the ambient character variety over the full ring of integers $\mathcal O_L$; it does not require membership in a strict locally closed integral boundary model at every finite prime. Ask also for the version fixing only the boundary characteristic polynomials with roots of unity, and for density in the entire character variety without boundary conditions. In the latter version take $K=\mathbb Q$. The same extension must work throughout the selected locus and all its components.

Short English statement

Are integral characters of determinant-one local systems Zariski dense in every component of a curve character variety after one finite number-field extension, also when quasi-unipotent boundary data are fixed?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2776-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2776

This question uses the source's determinant-one $\mathrm{SL}_r$ character variety and ambient integrality convention. Its theorem gives one finite number-field extension with dense integral points for every smooth complex curve and rank, for exact quasi-unipotent boundary classes, for their characteristic polynomials, and without boundary conditions. It imposes no genus or Jordan-form restriction. The theorem does not assert the corresponding statement for unrestricted $\mathrm{GL}_r$ representations.

Integral-point density on curve character varieties

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 027; 2026-09-25), the asserted result is that every component of the $\mathrm{SL}_r$ character variety of a smooth complex curve has Zariski-dense integral points after one finite field extension, including prescribed exact quasi-unipotent boundary classes.

Mathematical statement of the claimed result

Manuscript 1.
Take $r\ge1$ and an algebraic curve $X/\mathbb C$ that is connected and smooth. Let $M_r(X)$ be the natural integral $\operatorname{SL}_r$ character scheme. At the punctures of a smooth projective completion, prescribe quasi-unipotent conjugacy classes $C_i\subset\operatorname{SL}_r(\mathbb C)$ and let $Y$ be their reduced relative locus of semisimple representations. Choose a number field $K\subset\mathbb C$ containing all boundary eigenvalues. The claim gives one finite number-field extension $L/K$ such that \[ \overline{M_r(X)(\mathcal O_L)\cap Y(\mathbb C)}^{\rm Zar}=Y. \] Integrality means extension to the ambient character scheme over the full integer ring, not avoidance of smaller Jordan strata at all reductions. Prescribing only boundary characteristic polynomials with root-of-unity roots, or no boundary data, also works. For the relative cases and for projective $X$, distinct primes $p,q>r$ permit \[ F=K(\sqrt2,\zeta_{pq}),\qquad L=F(\zeta_r,\{u^{1/r}:u\in\mathcal O_F^{\times}\}). \] Without boundary data one may start with $K=\mathbb Q$.

Outline of the argument

For manuscript 1 (family 027; 2026-09-25), filtered local systems are cut along seams and glued with integral splittings. Rank induction handles closed and nonclosed facets together, while flag comparisons recover exact nonsemisimple boundary data rather than only its semisimplification.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 027; 2026-09-25).

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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