1 · The essentials
The question is whether certain spaces built from a curve contain enough points described by algebraic integers to spread throughout those spaces. The manuscript answers this for every smooth, connected complex algebraic curve and every matrix size, when the matrices have determinant one. After choosing a suitable number field, points with coordinates in its ring of integers are dense in a precise algebraic sense. This means they cannot all be trapped in a smaller algebraic closed subset of any component of the space. The result also permits specified behavior around the curve’s boundary, subject to the restrictions described below. A key distinction is that integer-valued coordinates are measured in the surrounding space, while the exact boundary requirements are imposed on the complex representation. The main result has a Lean formalization.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a building with several rooms, where you want marked locations to escape every smaller region that can be drawn using a restricted set of rules. Enlarging the number field is like enlarging the permitted list of location labels until the marked locations meet that requirement in every room. This image does not capture algebraic density precisely, which is different from being close to every location in the ordinary sense.
Why it matters
The repository describes this as resolving the determinant-one curve case of Litt’s integral-density question. Its scope includes every rank and every component, rather than only a selected matrix size or part of the space. Allowing exact nonsemisimple boundary conjugacy classes is also explicitly part of the result.
2 · Going further
Let be a smooth connected complex algebraic curve and let be a rank. The main manuscript establishes potential Zariski density of integral points on its character variety, the space recording representations by determinant-one matrices up to the identifications encoded by the character variety. Here “potential” means that one may pass to a suitable number field. Over the full ring of integers of one finite extension , the integral points are Zariski dense in every component: they are not contained in a proper Zariski-closed subset of that component.
The result allows prescribed quasi-unipotent boundary monodromy, including exact nonsemisimple conjugacy classes. These are restrictions on the matrices describing behavior around the boundary. The abstract makes an important distinction: integrality is measured in the ambient character variety, whereas the exact boundary conditions are imposed on the complex representation. The statement therefore keeps the arithmetic condition on the character-variety point separate from the precise boundary condition on the representation.
The repository places this result within Litt’s integral-density question and describes it as resolving the determinant-one curve case. The supplied abstract states the result and its scope but does not describe the argument’s mechanism, so no proof strategy can be extracted from this source. Only one manuscript is supplied; no relationship with companion manuscripts is specified. The main result is identified as Lean-formalized, without a more detailed account of the formalization’s coverage.
Useful notions
- Smooth complex algebraic curves
- Matrix representations and determinant one
- Character varieties
- Number fields and their rings of integers
- Zariski density
- Boundary monodromy
What was known before / what is new
Before
The repository places the work in the context of Litt’s integral-density question. The supplied text gives no earlier partial results or historical chronology, so it does not support a more detailed account of what was previously known.
New
The main result gives potential Zariski density for determinant-one character varieties of smooth connected complex curves in every rank and on every component. It uses the full ring of integers of one number field. It also allows prescribed quasi-unipotent boundary conjugacy classes, including exact nonsemisimple classes.
Limits and assumptions
The stated hypotheses concern smooth connected complex algebraic curves and determinant-one representations. Prescribed boundary classes must be quasi-unipotent, though they need not be semisimple. The number field may need to be enlarged; the source does not assert density over an arbitrary field fixed in advance. Integrality is an ambient character-variety condition, not a separate assertion that every matrix entry of the chosen complex representation is integral. The main result has a Lean formalization, but the supplied text does not specify its coverage in finer detail. No conclusions beyond the determinant-one curve case, or specific remaining open cases, are supplied.
Mini glossary
- Character variety —
- An algebraic space recording matrix representations through the identifications encoded by their characters.
- Zariski density —
- A set is Zariski dense if no proper algebraically closed subset of the space contains it.
- Ring of integers —
- The ring of algebraic integers within a number field.
- Potential Zariski density —
- Zariski density that holds after passing to a suitable finite extension of the number field.
- Boundary monodromy —
- The matrix behavior assigned by a representation to loops around the curve’s boundary.
- Quasi-unipotent —
- A matrix is quasi-unipotent if some positive power of it is unipotent, meaning that power differs from the identity by a nilpotent matrix.
Keywords : Character varieties · Integral points · Potential Zariski density
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Integral points on character varieties of curves
25/09/2026We 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.