The Minimum Index of a Non-Congruence Subgroup of Sl2 Over an Arithmetic Domain. Ii: The Rank Zero Cases

Abstract

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 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.

Non-standard automorphisms and non-congruence subgroups of SL2 over Dedekind domains contained in function fields

“…We combine all the results of this section together with the explicit values of ncs(C) given in [12]. Part (b) is proved in a similar way.…”

“…(b) We adopt the notation used in [12,Section 5]. Note that the groups ( ) in that paper are different from the groups (a) in the present paper.…”

“…It turns out that in nearly all cases this index is greater than ncs(C). In [10][11][12] we determine ncs(C). In most cases it turns out to be 2.…”

“…In previous papers the authors [10][11][12] have determined ncs(C), the minimal index of a non-congruence subgroup of . Now we use non-standard automorphisms to determine the minimal index of a non-congruence subgroup of prescribed level, in particular those of level zero and level C, the extremal cases.…”

Five wild nearfields

“…where Ä SL 2 R is the free product of p cyclic groups of order p C 1; see [13,Theorem 5.3], which relies on results in [21], or [20, Section 2.4.4, p. 115 and Exercise 3, p. 117 and Exercise 3, p. 120]. The free factor cannot contain the central involution id, hence identifying with its image in G we obtain…”