The cusp amplitudes and quasi - level of a congruence subgroup of SL 2 over any Dedekind domain

This is the latest part of an ongoing project aimed at extending algebraic properties of the classical modular group SL2(Z) to equivalent groups in the theory of Drinfeld modules. We are especially interested in those properties which are important in the classical theory of modular forms. Our results are intended to be applicable to the theory of Drinfeld modular curves and forms. Here we are concerned with the cusp amplitudes and level of a subgroup of such a group (in particular a congruence subgroup). In the process we have discovered that most of the theory of congruence subgroups, including the properties of their cusp amplitudes and level, can be extended to SL2(D), where D is any Dedekind ring. This means that this theory can be extended to a non-arithmetic setting. We begin with an ideal theoretic definition of the cusp amplitudes of a subgroup H of SL2(D) and extend the remarkable results of Larcher for the congruence subgroups of SL2(Z). We then extend the definition of the cusp amplitude and level of a subgroup H of SL2(D) by introducing the notions of quasi-amplitude and quasi-level. Quasi-amplitudes and quasi-level encode more information about H since they are not required to be ideals. In general although the level and quasi-level can be very different, we show that for many congruence subgroups they are equal. As a bonus our results provide several new necessary conditions for a subgroup of SL2(D) to be a congruence subgroup. These include an inequality between the index and level of a congruence subgroup.