On Non-Congruence Subgroups of the Analogue of the Modular Group in Characteristic p

Let k[t] be the polynomial ring over a finite field k. The group SL2(k[t]) is often referred to as the analogue, in characteristic p, of the classical modular group SL2(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{Z}$$ \end{document}), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{Z}$$ \end{document} is the ring of rational integers. It is well-known that the smallest index of a non-congruence subgroup of SL2(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{Z}$$ \end{document}) is 7. Here we compute this index for SL2(k[t]). (In all but 6 cases it turns out to be 1 + q, where q is the order of k.)