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

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

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