Browse › Number theory › 027 Potential integral density on curve character varieties
Integral points on character varieties of curves
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Abstract
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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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Potential integral density on curve character varieties (Lean scope notes) (lean/docs/027.md, 2026). The Lean statement covers only a supporting result: under the stated diagram and existing-solution hypotheses, the marked values produced by the marked-surface construction satisfy every seam equation; potential integral density, prescribed-boundary density and the all-component conclusions lie outside its scope. [family-level]
- OpenAI math release, CharacterVarietiesAllSeamsSupport.lean (lean/ComparatorChallenges, 2026). This comparator file states the theorem
producedMarkedValues_allSeams, the seam-compatibility statement referenced in the scope note.
Related prior literature mentioned by others
- Daniel Litt, Problems I Like, Problem 7 (last edited 2026-05-08). This is the potential-integral-density question whose determinant-one curve case the manuscript addresses, with remarks connecting the broader question to integrality of rigid local systems and isomonodromy foliations.
- Simone Coccia and Daniel Litt, Density of integral points in the Betti moduli of quasi-projective varieties (arXiv:2507.00167, 2025-06-30). Highlighted in Litt's Problem 7 remarks, this rank-two predecessor proves potential integral density for relative SL₂ character varieties with algebraic-integer boundary traces using reduction to curves and mapping-class-group orbits.
Possibly relevant literature (found by the agent, not checked)
- T. Hausel, F. Rodriguez-Villegas (appendix by N. M. Katz), Mixed Hodge polynomials of character varieties (arXiv:math/0612668, Invent. Math., 2006). It counts points of twisted GL_n character varieties of closed surfaces over finite fields using the character table of GL_n(F_q), which may relate to the paper's finite-field proof that the twisted surface equation is irreducible.
- T. Hausel, E. Letellier, F. Rodriguez-Villegas, Arithmetic harmonic analysis on character and quiver varieties (arXiv:0810.2076, Duke Math. J. 160 (2011)). It extends the finite-field character-sum counts to punctured surfaces with generic semisimple boundary classes, which may bear on the paper's point-count approach to irreducibility with prescribed boundary data.
- M. Kamgarpour, G. Nam, A. Puskás, Arithmetic geometry of character varieties with regular monodromy (arXiv:2209.02171, 2022). It proves smoothness and polynomial point counts, including the number of components, for character varieties with regular semisimple and regular unipotent monodromy, which may relate to the paper's handling of nonsemisimple quasi-unipotent boundary classes and its all-components conclusion.
- J. P. Whang, Arithmetic of curves on moduli of local systems (arXiv:1803.04583, Algebra Number Theory 14 (2020)). It studies integral points on curves in rank-two relative character varieties with fixed boundary traces, complementing the Whang descent paper the manuscript cites for rank-two integral characters.
Digests and expositions
- Gradient Damage, Large Prime Gaps Have Positive Density, Artin's Primitive Root Conjecture and 8 More (video exposition, 2026-10-14). The chapter beginning at 2:27 illustrates this paper's topic through traces of SL₂ representations of a once-punctured torus and the Markoff equation, presenting the result as OpenAI's claim rather than a proof verification.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 027
Independent sites; their content is not reviewed here.
Cite
@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}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/027.md. The Lean statements have not been checked by people against the paper's claims.
Family 027: 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.
This is the only paper in this family.