Result 027, Number theory

Potential integral density on curve character varieties

Resolves the determinant-one curve case of Litt's integral-density question. For every smooth connected complex algebraic curve and every rank, integral points become Zariski dense in every component of its SLr character variety over the full ring of integers of one number field. Prescribed quasi-unipotent boundary conjugacy classes are allowed, including nonsemisimple classes.

Proof

The bigger picture

Why it matters

Matrix descriptions of loops on an algebraic curve form geometric spaces of their own. The manuscript claims that arithmetic points are abundant throughout these spaces, not trapped in smaller algebraic subsets, in the determinant-one setting.

What changes?

The manuscript reports that, for every smooth connected complex algebraic curve and every rank r, integral points are Zariski dense in every component of its SLr character variety, over the full ring of integers of one number field. This variety records determinant-one matrix representations of the curve's loops. Prescribed boundary conjugacy classes are allowed when their eigenvalues are roots of unity, even when the matrices are not diagonalizable.

What does that help mathematicians do?

Density lets researchers test polynomial identities on integral points: an identity valid on all of them must hold throughout the component. It also guarantees integral points outside any proper algebraic exceptional subset. The scope is important: integrality is measured in the ambient character variety, while the exact boundary conjugacy conditions are imposed on the complex representation. Thus the claim includes fixed non-diagonalizable boundary behavior, not just its eigenvalue data.

Are there practical applications?

The immediate value is foundational in number theory: it links the geometry of curve representation spaces to points with algebraic-integer coordinates. It answers Litt's integral-density question for determinant-one representations of curves, without asserting a result for other groups or higher-dimensional varieties.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Integral points on character varieties of curves

September 25, 2026 27 pages

We prove potential Zariski density of integral points on SLr-character varieties of smooth complex curves in every rank. The result allows prescribed quasi-unipotent boundary monodromy, including exact nonsemisimple conjugacy classes, and holds on every component over the full ring of integers of one finite extension. Here integrality is measured in the ambient character variety, while the exact boundary conditions are imposed on the complex representation.

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Cite (BibTeX) 
@misc{OAI:Integral-points-on-character-varieties-of-curves-September-25-2026,
  author = {{OpenAI}},
  title = {{Integral points on character varieties of curves}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Integral-points-on-character-varieties-of-curves-September-25-2026/paper.pdf}{OAI:Integral-points-on-character-varieties-of-curves-September-25-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.