Action of the mobius group ¨ M = hx, y : x 2 = y 6 = 1i on certain real quadratic fields

Abstract

Let C 0 = C ∪ {∞} be the extended complex plane and M = ­ x, y : x 2 = y 6 = 1® , where x(z) = −1 3z and y(z) = −1 3(z+1) are the linear fractional transformations from C 0 → C 0 . Let m be a squarefree positive integer. Then Q∗ ( √ n) = { a+ √ n c : a, c 6= 0, b = a 2−n c ∈ Z and (a, b, c) = 1} where n = k 2m, is a proper subset of Q( √ m) for all k ∈ N. For non-square n = 3h Qr i=1 p ki i , it was proved in an earlier paper by the same authors that the set Q 000 ( √ n) = { α t : α ∈ Q∗ ( √ n), t = 1, 3} is M-set ∀ h ≥ 0 whereas if h = 0 or 1, then Q∗∗∗√ n) = { a+ √ n c : a+ √ n c ∈ Q∗ ( √ n) and 3 | c} is an M-subset of Q 000 ( √ n) = Q∗ ( √ n) ∪ Q∗∗∗( √ 9n). In this paper we prove that if h ≥ 2, then Q 000 ( √ n) = (Q∗ ( pn 9 )\Q∗∗∗( pn 9 ))∪Q∗ ( √ n)∪Q∗∗∗( √ 9n) and also determine its proper M-subsets. In particular Q( √ m) \ Q = ∪Q 000 ( √ k 2m) for all k ∈ N.