On Ramanujan's cubic continued fraction, II

Abstract

We show that all the periodic points with period $n \ge 3$ of the algebraic function defined by the relation $g(x,y) = x^2+2xy^2-y = 0$ are expressible in terms of values of Ramanujan's cubic continued fraction $c(τ)$. It is shown that the values $c(w/2), c(w/6), c(w/6+1/2)$ are periodic points, for specific elements $w$ in the order of discriminant $-d$ in $K = \mathbb{Q}(\sqrt{-d})$, with $-d \equiv 1$ (mod $8$); and all the periodic points are conjugates of these values or specific linear fractional expressions in their conjugates, times cube roots of unity. These periodic points are determined by relating them to the periodic points of the algebraic function defined by $k(x,y)= -xy^2+x^2+y^2+3x = 0$; and these periodic points (for periods $\ge 2$) generate ring class fields over $K$.