The order and level of a subgroup of GL2 over a Dedekind ring of arithmetic type

Abstract

SynopsisLet R be a commutative ring and let q be an R-ideal. Let En(R) be the subgroup of GLn(R) generated by the elementary matrices and let En(R, q) be the normal subgroup of En(R) generated by the q-elementary matrices. For each subgroup S of GLn(R) the order of S, o(S), is the R-ideal generated by xij, xii − xjj (i ≠ j), where (xij) ∈ S, and the level of S, l(S), is the largest R-ideal q0 with the property that En (R, q0) ≦ S. It is known that when n ≧ 3, the subgroup S is normalised by En(R) if and only if o(S) = l(S). It is also known that this result does not hold when n = 2. For example, there are uncountably many normal subgroups S of SL2(ℤ) such that o(S) ≠ {0} and l(S) = {0}, where ℤ is the ring of integers. In this paper we prove that, when A is a Dedekind ring of arithmetic type containing infinitely many units, the order q and level q′ of a subgroup S of GL2(A), normalised by E2(A), are closely related. It is proved that Ψ(q)≦q′, where ≦(q) = 12uq, with u the A-ideal generated by u2 − 1 (u ∈ A*), when A is contained in a number field, and Ψ(q) = q3, when A is contained in a function field.