Density of integral points in the Betti moduli of quasi-projective varieties

2025/06/30 by Simone Coccia, Daniel Litt, Coccia, Simone +1 · 1 voice
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometry and complex manifolds #Number Theory (math.NT) #math.AG #math.GT #math.NT

paper · pdf · doi:10.48550/arxiv.2507.00167

openalex publication_date 2025/06/30 · arxiv published 2025/06/30 · arxiv updated 2025/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Y be a smooth quasi-projective complex variety equipped with a simple normal crossings compactification. We show that integral points are potentially dense in the (relative) character varieties parametrizing SL₂-local systems on Y with fixed algebraic integer traces along the boundary components. The proof proceeds by using work of Corlette-Simpson to reduce to the case of Riemann surfaces, where we produce an integral point with Zariski-dense orbit under the mapping class group.

Citations

• A new proof of Chen's theorem for Markoff graphs
• Algebraicity and integrality of solutions to differential equations
• Motives, mapping class groups, and monodromy
• Finite braid group orbits on SL₂-character varieties
• Morphisms of character varieties
• Integrality of the Betti moduli space
• Canonical representations of surface groups
• Coordinate rings of some SL₂-character varieties
• Nonabelian level structures, Nielsen equivalence, and Markoff triples
• Cohomologically rigid local systems and integrality
• Nonlinear descent on moduli of local systems
• Markoff Triples and Strong Approximation
• Representations of quasiprojective groups, Flat connections and Transversely projective foliations
• Harmonic Maps to Buildings and Singular Perturbation Theory
• On a conjecture of Deligne
• Topological Dynamics on Moduli Spaces II
• Topological Dynamics on Moduli Spaces, I

Discussions

• You can read the paper here: arxiv.org/abs/2507.00167 It even has some pictures. [bsky, 3 points, 0 comments]

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