• Journal of the Korean Mathematical Society
• /
• v.55 no.2
• /
• pp.343-372
• /
• 2018

Let K be an imaginary quadratic field with ring of integers ${\mathcal{O}}_K$. Let E be an elliptic curve with complex multiplication by ${\mathcal{O}}_K$, and let $h_E$ be the Weber function on E. Let $N{\in}\{2,3,4,6\}$. We show that $h_E$ alone when evaluated at a certain N-torsion point on E generates the ray class field of K modulo $N{\mathcal{O}}_K$. This would be a partial answer to the question raised by Hasse and Ramachandra.

• Honam Mathematical Journal
• /
• v.31 no.4
• /
• pp.463-478
• /
• 2009

Let k be an imaginary quadratic field, ℌ the complex upper half plane, and let ${\tau}{\in}k{\cap}$ℌ, q = exp(${\pi}i{\tau}$). And let n, t be positive integers with $1{\leq}t{\leq}n-1$. Then $q^{{\frac{n}{12}}-{\frac{t}{2}}+{\frac{t^2}{2n}}}{\prod}^{\infty}_{m=1}(1-q^{nm-t})(1-q^{nm-(n-t)})$ is an algebraic number [10]. As a generalization of this result, we find several infinite series and products giving algebraic numbers using Ramanujan's $_{1{\psi}1}$ summation. These are also related to Rogers-Ramanujan continued fractions.