THE MINIMUM INDEX OF A NON-CONGRUENCE SUBGROUP OF SL2 OVER AN ARITHMETIC DOMAIN. II: THE RANK ZERO CASES

Let K be a function field of genus g with a finite constant field Fq . 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 SL2(C). It turns out that this index is always equal to the minimum index of an arbitrary proper subgroup in SL2(C). The minimum index of a normal non-congruence subgroup is also determined.