G-subsets of an invariant subset ℚ*(√k 2m) of ℚ(√m) \ ℚ under the modular group action

Abstract

PSL₂(Z) is the well known modular group with the presentation(x, y: x² = y³ = 1) where x: C' → C' and y: C' → C' are the Möbius transformations defined by: x(z) =-1/z, y(z) = x-1/xLet n = k²m, where m is a square free positive integer and k is any non zero integer. Then ℚ^*(√n) = {a+√n/c: a, c ≠0, b = a²-n/c ∈ Z and(a, b, c) = 1} is a G-subset of ℚ(√m)\Q. In this paper we are interested in finding the cardinality of the set E_p^r, r ≥ 1, consisting of all classes [a, b, c](mod p^r) of the elements of Q^*(√n). Also we determine, for each non-square n, the all G-subsets of Q ^*(√n) under the modular group action by using classes [a, b, c](mod n).