The Minimum Index of a Non‐Congruence Subgroup of SL 2 Over an Arithmetic Domain. II: The Rank Zero Cases

Let K be a function field of genus g with a finite constant field F q . Choose a place ∞ of K of degree δ and let C be the arithmetic Dedekind domain consisting of all elements of K that are integral outside ∞. An explicit formula is given (in terms of q, g and δ) for the minimum index of a non‐congruence subgroup in SL 2 ( C ). It turns out that this index is always equal to the minimum index of an arbitrary proper subgroup in SL 2 ( C ). The minimum index of a normal non‐congruence subgroup is also determined.