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Litt's potential integral density question for SLr\mathrm{SL}_r character varieties of curves

For a smooth complex variety XX, the SLr\mathrm{SL}_r-character variety of π1(X)\pi_1(X) has a natural model Mr(X)M_r(X) over Z\mathbb Z. Litt asked whether integral points on such character varieties become Zariski dense after a finite extension of the ground field, including a variant with prescribed boundary monodromy, and whether one number field LL suffices so that the points over the full ring of integers OL\mathcal O_L are dense in every component. Coccia and Litt proved this for SL2\mathrm{SL}_2 and PGL2\mathrm{PGL}_2 and conjectured it for Chevalley groups with fixed boundary data. For a smooth connected curve XX, every rank rr and prescribed quasi-unipotent boundary classes, is there one number field over whose full ring of integers the integral points are Zariski dense?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Construction
Field
Arithmetic geometry; character varieties, integral points
Posed by
Daniel Litt (Problems I Like, Problem 7; survey 2024, Question 5.4.3(2)); Simone Coccia and Daniel Litt (2025, Conjecture 1.1.1)
Year posed
2024
Years open
2y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
15 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every smooth connected complex curve XX, rank rr and quasi-unipotent conjugacy classes at the punctures (nonsemisimple allowed), there is a finite extension LL of the field of eigenvalues with Mr(X)(OL)∩Y(C)M_r(X)(\mathcal O_L)\cap Y(\mathbb C) Zariski dense in every component of the relative locus YY; an explicit LL is given. The same holds when only characteristic polynomials are fixed, and for the whole character variety with base field Q\mathbb Q. This settles the determinant-one curve case of Litt's question and the SLr\mathrm{SL}_r curve case of the Coccia-Litt conjecture for quasi-unipotent data; higher-dimensional varieties are not treated.

What the AI did

The release README says all results were produced by an unreleased internal OpenAI model using one fixed procedure, about three hours of ChatGPT Pro thinking compute per result on average. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Litt's question as the paper states it; Litt's sources were not opened. The family has no Lean formalization (lean/docs/027.md does not exist at the pinned commit). Integrality is ambient (points of Mr(X)M_r(X) over OL\mathcal O_L whose complex representation has the prescribed classes), which the paper distinguishes from integral points of a locally closed integral model. Varieties of dimension above one are outside the result, as the paper says.

Sources

Changelog1 change

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