On the modular functions arising from the Theta constants

Abstract

Some modular functions arising from the theta constants $ \vartheta_2(\tau)$, $ \vartheta_3(\tau)$, $ \vartheta_4(\tau)$ are investigated. Let $n$ be an odd square-free positive integer as in [4,7]. It is obtained a necessary and sufficient condition that $ \varphi_{\delta,\rho,3}(\tau)=\prod_{\delta|n,\rho|n}\Big({\vartheta_3(\delta\tau) \over\vartheta_3(\rho\tau)}\Big)^{r_\delta}$  is invariant with respect to transformations in $ \theta(n)$. Also, It is deduced that $ \varphi_{\delta,\rho,i}(\tau)$ is a modular function on $ P^{-2}\theta(n)P^2$, $ \theta(n)$, $P^{-1}\theta(n)P$, for $ i=2,3,4$, respectively. Thus, the result of L. Wilson's paper [7] is generalized. Furthermore, let $ m$ and $ n$ denote positive integers. Let $ r$, $ r_1$, $ r_2$ be integers such that $ r(m-1)(n+1)\equiv 0({\rm mod}~8)$, $ r_1(m-1)(n-1)\equiv 0({\rm mod}~ 8)$, $ r_2^2(n-m)(nm-1)\equiv 0({\rm mod}~8)$, it is shown that $ T_{m,n,i}^r(\tau)=\Big({\disp{\vartheta_i(\tau)\vartheta_i(n\tau)\over\vartheta_i(m\tau) \vartheta_i (mn\tau)}}\Big)^r$, $ H_{m,n,i}^{r_1}(\tau)=\Big({\disp{\vartheta_i(m\tau)\vartheta_i(n\tau)\over \vartheta_i(\tau)\vartheta_i(mn\tau)}}\Big)^{r_1}$ and $ \Phi_{m,n,i}^{r_2}(\tau)=\Big({\disp{\vartheta_i(m\tau)\over\vartheta_i(n\tau)}\Big)^{ r_2 }} $ are modular functions on $ \theta(mn)$, when $ i=3$. Similar results are deduced for $ P^{-2}\theta(mn)P^2$ and $ P^{-1}\theta(mn)P$, the suffixes 3 being replaced by 2 and 4, respectively. Therefore, the modular functions used in B. C. Berndt's paper [1] is rewritten for theta constants.