View source for Potential integral density on curve character varieties

'''Potential integral density on curve character varieties''' asks whether, for a smooth complex algebraic curve, the integral points of its SL<sub>''r''</sub>-character variety become Zariski dense after one finite extension of the ground field, also when the conjugacy classes of the monodromy around the punctures are prescribed. A manuscript in OpenAI's public mathematics collection, dated 25 September 2026, claims this in every rank for quasi-unipotent boundary classes. The collection's Lean formalization covers one supporting step of the construction, not the main theorem.

== Background ==
* As recounted in the manuscript's introduction:
** ''Integrality of local systems.'' Simpson conjectured that rigid irreducible representations of fundamental groups of smooth projective varieties are defined over rings of integers. Esnault and Groechenig proved integrality for irreducible cohomologically rigid local systems with finite determinant and quasi-unipotent boundary monodromy, and de Jong and Esnault proved that a suitable complex local system implies absolutely irreducible integral ℓ-adic local systems for every prime ℓ; the manuscript stresses that this prime-by-prime existence differs from density over the ring of integers of one number field.
** ''The density question.'' Litt asked for potential integral density on character varieties, with a boundary variant, and his survey asks whether one number field suffices. Coccia and Litt formulated a related conjecture for Chevalley groups with fixed boundary data and proved potential integral density for SL<sub>2</sub> and PGL<sub>2</sub>, using integral representations and mapping-class-group dynamics.
** ''Related work.'' Whang developed a descent framework for rank-two integral characters with fixed boundary traces, and Ghosh and Sarnak studied integral points on the Markoff cubic surfaces, including infinitely many admissible parameters without integral points. Mellit and Su gave geometric decompositions of character varieties with generic semisimple boundary data.

== The original result ==
* '''Claim, 25 September 2026.''' The main theorem of ''Integral points on character varieties of curves'' states that for a smooth connected complex algebraic curve, every rank ''r'' and prescribed quasi-unipotent conjugacy classes at the punctures, there is a finite extension ''L'' of a number field containing their eigenvalues such that the points of the integral model of the SL<sub>''r''</sub>-character variety over the ring of integers of ''L'' are Zariski dense in the locus with the prescribed classes, and hence in each of its irreducible components. For projective curves and for prescribed boundary classes the field is explicit: it adjoins √2 and a primitive ''pq''-th root of unity, for distinct primes ''p'', ''q'' larger than ''r'', and then a primitive ''r''-th root of unity and the ''r''-th roots of all units of the resulting field.
* ''Scope, as stated:'' integrality is ambient: an integral point extends to the integral model of the whole character variety and has the prescribed classes over the complex numbers, but need not avoid smaller Jordan classes after reduction at every prime. The manuscript states that it answers the determinant-one form of Litt's question for curves in every rank and proves the SL<sub>''r''</sub> curve case of the Coccia–Litt conjecture for quasi-unipotent boundary data; the higher-dimensional, higher-rank question remains outside its scope. Without boundary conditions on a nonprojective curve, density already holds over the integers.
* ''Method, as described:'' the boundary matrices are capped by disks carrying scalar local systems on the layers of invariant flags, giving diagrams of surfaces of several ranks glued along flags. Integral density is proved for all such diagrams by induction on the largest rank, cutting surfaces into disks and recovering the lost extension data by linear choices. Closed surfaces are cut along one nonseparating circle, and powers of a Dehn twist enlarge the closure of the integral points; irreducibility of the twisted surface equation is proved by counting over finite fields, following Liebeck and Shalev, with the Lang–Weil estimate. Dirichlet's unit theorem gives the single field, and the exact Jordan classes are selected last on the character variety.
* '''Sources.''' Commit <code>adc7f12</code> of the openai/math repository: [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integral-points-on-character-varieties-of-curves-September-25-2026 ''Integral points on character varieties of curves''].
* '''Formalization.''' The [https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/lean/docs/027.md scope note], added to the repository on 8 October 2026 and revised on 26 October 2026, states that the Lean development proves a supporting compatibility result for the manuscript's marked-surface construction: given a proper diagram, a suitable seam, puncture data and an existing solution over a field, the produced marked values satisfy every seam equation. It states that existence of integral points and Zariski density are not established, and that the main theorem is outside the comparison. Comparator statement: [https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/lean/ComparatorChallenges/CharacterVarietiesAllSeamsSupport.lean <code>CharacterVarietiesAllSeamsSupport.lean</code>].

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 31 October 2026)
|-
| 2026-10-10 || [https://github.com/mathvet/mathvet MathVet], a GitHub review of openai/math, tables at repository commit <code>c76f1e4</code> of 14 October 2026 || Review, fidelity of the formal statement || Rates this family "supporting-only": the Lean states only a compatibility lemma inside the cutting construction and says nothing about integral points, Zariski density, number fields or rings of integers, SL<sub>''r''</sub> character varieties or quasi-unipotent boundary classes. Among the definitions to check it notes that quasi-unipotence is not imposed and that the solutions are field-valued points, not integral points. It records a sandboxed Comparator pass on 10 October 2026, with about 27,400 lines in the dependency cone. || Review repository. Read on 31 October 2026.
|-
| 2026-10-10 || [https://github.com/carlostoledo1891/cert-machine/blob/e35cd57a5fbd5b870c66d88697297a33f64f0eb2/certs/openai-math-kernel.json cert-machine], openai/math lane K, run 38019899278 || Verification, two kernels || Records the challenge <code>CharacterVarietiesAllSeamsSupport</code> as certified: Lean's default kernel and the nanoda kernel both accept the solution, in 3,029 s. C. Toledo's [https://carlostoledo.co/reports/openai-math.html report] lists the lane's results. || Repository. The run concerns the formalized statement only.
|-
| 2026-10-10 to 2026-10-14 || [https://github.com/leanprover/con-leche/blob/90577d9e49ee3aed45da3d421bdf5dc780192304/DESIGN.md con-leche development journal], task #348 || Verification, second kernel || In its run over all Comparator challenges of the collection at commit <code>fd4aeeb</code>, the journal's table of exceptions does not list this family's challenge, so it was accepted by con-leche and by Lean's official kernel. || Development journal.
|-
| 2026-10-12 || ''AI Proved What?'', episode 027 [https://aiprovedwhat.com/episodes/027-potential-integral-density-curve-character/ Integral points, everywhere?], remade version of 25 October 2026 || Reading, exposition || Video explainer of the 27-page preprint, starting from the rank-two character variety of the once-punctured torus with coordinates given by traces. It recalls the work of Simpson, Esnault and Groechenig, de Jong and Esnault, Litt, Coccia and Litt, and Ghosh and Sarnak, and states that integrality is measured in the full character variety, that higher-dimensional varieties in rank three and up are not covered, and that density without conditions at the punctures was already easy over the integers. It also reports 13 further Lean files of the collection not linked from its catalog, with supporting steps of the same construction and the finiteness of the field obtained by adjoining roots of units. || Episode status, last checked 30 October 2026: posted as a preprint; under expert review, not yet; peer reviewed, not yet; checked in Lean, no, only supporting steps, which the site did not run; no public response found. The site states that it is made with Anthropic's Claude. Read on 31 October 2026.
|}

== Notes ==
* The background summarizes the manuscript's own account of earlier work.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]