Abstract We study the modularity of Ramanujan’s function k ( τ ) = r ( τ ) r 2 ( 2 τ ) k(\tau )=r(\tau ){r}^{2}(2\tau ) , where r ( τ ) r(\tau ) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k ( τ ) k(\tau ) of “an” level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ \tau in an imaginary quadratic field, the value k ( τ ) k(\tau ) generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ 1 ( 10 ) {{\mathrm{\Gamma}}}_{1}(10) . Furthermore, we suggest a rather optimal way of evaluating the singular values of k ( τ ) k(\tau ) using the modular equations in the following two ways: one is that if j ( τ ) j(\tau ) is the elliptic modular function, then one can explicitly evaluate the value k ( τ ) k(\tau ) , and the other is that once the value k ( τ ) k(\tau ) is given, we can obtain the value k ( r τ ) k(r\tau ) for any positive rational number r immediately.

中文翻译：

---

拉马努金函数 k(τ)=r(τ)r2(2τ) 及其模块性

摘要 我们研究了 Ramanujan 函数 k ( τ ) = r ( τ ) r 2 ( 2 τ ) k(\tau )=r(\tau ){r}^{2}(2\tau ) 的模块性，其中 r ( τ ) r(\tau ) 是 Rogers-Ramanujan 连分数。我们首先找到“an”级k(τ)k(\tau)的模方程，得到模方程满足的一些对称关系和一些同余关系；这些关系对于减少寻找模方程的计算成本非常有用。我们还表明，对于虚二次场中的某些 τ \tau，值 k ( τ ) k(\tau ) 在虚二次场模 10 上生成射线类场；这是因为函数 k 是 Γ 1 ( 10 ) {{\mathrm{\Gamma}}}_{1}(10) 上模函数域的生成器。此外，