On the cycles of indefinite quadratic forms and cycles of ideals II

Abstract

Let P and Q be two positive integers such that P < Q and let D = P-2 + Q(2) be a positive non- square integer. In the first section, we give some preliminaries from binary quadratic forms and quadratic ideals. In the second section, we show that given an ideal I = [Q, P + root D], there exists an indefinite symmetric quadratic form F-I = (Q, 2P,-Q) of discriminant 4 D which corresponds to I. We prove that I is always reduced, and so is F-I. Further, we prove that the cycle of F-I can be obtained using the cycle of I.