On Ramanujan's cubic continued fraction

Abstract

The periodic points of the algebraic function defined by the equation $g(x,y)=x^3(4y^2+2y+1)-y(y^2-y+1)$ are shown to be expressible in terms of Ramanujan's cubic continued fraction $c(\ddot{I}„)$ with arguments in an imaginary quadratic field in which the prime $3$ splits. If $w=(a+\sqrt{-d})/2$ lies in an order of conductor $f$ in $K$ and $9\mid N_{K/\mathbb{Q}}(w)$, then one of these periodic points is $c(w/3)$, which is shown to generate the ring class field of conductor $2f$ over $K$.

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