Modularity of certain products of the cubic continued fraction
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Let $C(\tau)$ be the Ramanujan's cubic continued fraction. We prove that under some condition on integers $a$ and $b\neq 0$, there are only finitely many functions of the form $C(\tau)^aC(2\tau)^b$ that generate the field of all modular functions on the congruence subgroup $\Gamma_0(12)$, partially extending the result of Park. We also establish certain modular and arithmetic properties of the function $t(\tau) = C(\tau)^{-2}C(2\tau)$. Our proofs employ the methods of Park and Gee, as well as some properties of $\eta$-quotients.
Abstract
Let $C(\tau)$ be the Ramanujan's cubic continued fraction. We prove that under some condition on integers $a$ and $b\neq 0$, there are only finitely many functions of the form $C(\tau)^aC(2\tau)^b$ that generate the field of all modular functions on the congruence subgroup $\Gamma_0(12)$, partially extending the result of Park. We also establish certain modular and arithmetic properties of the function $t(\tau) = C(\tau)^{-2}C(2\tau)$. Our proofs employ the methods of Park and Gee, as well as some properties of $\eta$-quotients.