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The minimum index of a non-congruence subgroup of \(\text{SL}_2\) over an arithmetic domain. (English) Zbl 1020.20033

Let \(A\) be an arithmetic Dedekind ring with only finitely many units. Thus, there are 3 possibilities: (1) \(A\) is the rational integers; (2) \(A\) is the ring of integers in an imaginary quadratic extension of the rational numbers; (3) \(A\) is the coordinate ring of an affine curve over a finite field with one point at infinity. Serre proved that there are subgroups of finite index in \(\text{SL}_2(A)\) which are not congruence subgroups. The paper concerns the least index \(\text{ncs}(A)\). It is well-known that \(\text{ncs}(A)=7\) in case (1). Grunewald-Schwermer proved that \(\text{ncs}(A)=2\) in case (2) with 4 exceptions. The present paper proves that \(\text{ncs}(A)=2\) in case (3) with a “small” set of exceptions. The determination of \(\text{ncs}(A)\) on the exceptional set is stated as an open problem.

MSC:

20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)
20E07 Subgroup theorems; subgroup growth
11F06 Structure of modular groups and generalizations; arithmetic groups
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