Cycles of Indefinite Quadratic Forms and Cycles of Ideals

Let ± denotes a real quadratic irrational integer with trace t = ± + ± and norm n = ±:±. Given a real quadratic irrational ° 2 Q(±), there are rational integers P and Q such that ° = P+± Q with Qj( ± + P)( ± + P). Hence for each ° = P+± Q, there is a corresponding ideal I ° = [Q;P + ±], and an inde¯nite quadratic form F°(x; y) = Q(x ¡ ±y)(x ¡ ±y) of discriminant t2 ¡ 4n. In this paper, we consider the cycles of I ° and cycles of F ° for some speci¯c values of ± = p D, where D 6 = 1 is a positive non-square integer.