Simpson’s integrality conjecture for rigid local systems (projective and quasi-projective cases)

Prove Simpson’s integrality conjecture asserting that rigid local systems are integral: specifically, determine whether integral points are Zariski-dense in the zero-dimensional components of the character variety X_{G,\underline{C}}(Y) for a smooth complex projective variety Y (and its quasi-projective variant), where G is a reductive (Chevalley) group and \underline{C} prescribes boundary conjugacy classes.

Background

The authors highlight Simpson’s conjecture as the antecedent to their main conjecture. In this context, the conjecture translates to the statement that isolated points (rigid local systems) of X_{G,\underline{C}}(Y) should be integral, equivalently that integral points are Zariski-dense in its zero-dimensional components. Despite significant partial progress via \ell-adic companions and results of Esnault–Groechenig and Klevdal–Patrikis under quasi-unipotent hypotheses, the general case remains unanswered.

Establishing this would resolve the fundamental integrality question for rigid local systems in both projective and quasi-projective settings, providing a cornerstone for broader nonabelian arithmetic geometry programs involving moduli of local systems.