, Volume 114, Issue 1, pp 205-220

On the concept of level for subgroups of SL2 over arithmetic rings

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Abstract

We define the concept of level for arbitrary subgroups Γ of finite index in the special linear group SL2(A S), whereA S is the ring ofS-integers of a global fieldk provided thatk is an algebraic number field, or card (S) ≥ 2. It is shown that this concept agrees with the usual notion of ‘Stufe’ for congruence subgroups. In the case SL2(O),O the ring of integers of an imaginary quadratic number field, this criterion for deciding whether or not an arbitrary subgroup of finite index is a congruence subgroup is used to determine the minimum of the indices of non-congruence subgroups.