Potential density of integral points in relative character varieties for Chevalley groups

Establish potential Zariski-density of integral points in the relative character variety X_{G,\underline{C}}(Y)_K for every Chevalley group scheme G over the integers, number field K with ring of integers \mathscr{O}_K, and boundary conjugacy classes \underline{C} in the adjoint quotient (G/_{\mathrm{ad}G})(\mathscr{O}_K)^n associated to a smooth quasi-projective complex variety Y equipped with a simple normal crossings compactification. Concretely, determine whether there exists a finite extension L/K such that the Zariski-closure of X_{G,\underline{C}}(Y)(\mathscr{O}_L) contains X_{G,\underline{C}}(Y)_K.

Background

The paper introduces a general conjecture asserting potential Zariski-density of integral points in relative character varieties X_{G,\underline{C}}(Y). Here Y is a smooth complex quasi-projective variety with a simple normal crossings compactification, G is a Chevalley group over \mathbb{Z}, and \underline{C} fixes boundary conjugacy classes via the adjoint quotient. The authors prove this conjecture for G = SL_2 and PGL_2, but the conjecture remains open in full generality for arbitrary Chevalley groups.

This problem extends Simpson’s integrality philosophy and links to expectations from Campana’s conjectures about log Calabi–Yau varieties admitting potentially dense sets of integral points; it also connects to cluster structure expectations for character varieties. A complete resolution would unify and generalize integrality phenomena across nonabelian moduli of local systems.