The cusp amplitudes and quasi-level of a congruence subgroup of \(\mathrm{SL}_{2}\) over any Dedekind domain (Q2909056)

One of the aims of this paper is to analyse various possible generalizations of the classical result that for a congruence subgroup of the modular group \(\mathrm{SL}(2,\mathbb{Z})\), the level is at most the index. The authors study \(\mathrm{SL}(2,D)\) over Dedekind domains \(D\). They introduce the finer notions of quasi-level and quasi-amplitude which contain more information although they have less structure (for instance, they are not ideals but only additive subgroups in general).NEWLINENEWLINENEWLINERecall that for a commutative ring \(R\) with unity, the level \(l(H)\) of a subgroup \(H\) of \(\mathrm{SL}(2,R)\) is the largest ideal \(q\) of \(R\) such that every conjugate of the unipotent upper triangular matrix with \((1,2)\)-th entry \(a\) is contained in \(H\) for every \(a \in q\). If one considers the additive subgroup \(\mathrm{ql}(H)\) of \(R\) consisting of all \(a \in R\) with the above property, it is an additive subgroup of \(R\), called the quasi-level; it is clear that the level is the largest ideal contained in \(ql(H)\). For the special case \(R = \mathbb{Z}\), both coincide. These are different in general; for instance, if \(R = k[T]\) for a finite field \(k\), \(\mathrm{SL}(2,R)\) has infinitely many finite index subgroups of level zero although the quasi-level is non-zero.NEWLINENEWLINENEWLINESimilarly, for a Dedekind domain \(D\), and a subgroup \(H\) of \(\mathrm{SL}(2,D)\), each cusp amplitude is a certain ideal associated to the cusps of \(H\). The authors consider all the conjugates of \(H\) and that gives a more general notion of quasi-amplitude of \(H\) which is only an additive subgroup of \(D\) in general; it contains the cusp amplitude as the largest ideal contained in it.NEWLINENEWLINENEWLINEThe first important result proved by the authors is the following generalization of a classical result for \(\mathrm{SL}(2,\mathbb{Z})\) due to Larcher:NEWLINENEWLINE Let \(D\) be a Dedekind domain and \(H\) be a subgroup of \(\mathrm{SL}(2,D)\). If \(A(H)\) denotes the set of all cusp amplitudes of \(H\), then for a congruence subgroup \(H\), the intersection of all elements of \(A(H)\) as well as the sum of all the elements of \(A(H)\) are both elements of \(A(H)\).NEWLINENEWLINE NEWLINEThere are various other interesting results in the paper although not all of them may look as striking as the one above. For instance, if the characteristic of \(D\) is at least \(5\), then the quasi-level and the level of a congruence subgroup of \(\mathrm{SL}(2,D)\) coincide. Using such results, the authors can draw some useful conclusions like:NEWLINENEWLINE NEWLINELet \(D\) be a Dedekind domain and \(q \neq 0\), an ideal. Suppose \(H\) is a subgroup of \(\mathrm{SL}(2,D)\) with level \(q\). If \(H\) is a congruence subgroup, then: (a) \(G(q) \leq H\); (b) the ideal sum of all cusp amplitudes of \(H\) is a cusp amplitude; (c) the intersection of all cusp amplitudes of \(H\) is a cusp amplitude; (d) the level \(q\) of \(H\) is a cusp amplitude of \(H\); (e) for any \(\alpha \in D\) which is invertible modulo \(q\), we have \(\alpha^2 \mathrm{ql}(H) \subseteq \mathrm{ql}(H)\).NEWLINENEWLINENEWLINESuch results and other variants here provide new necessary conditions for a subgroup to be a congruence subgroup. We can expect these ideas and results to be useful in the theory of Drinfeld modules.