Differential relations for modular forms of level five.
It is known that the solutions to the system of differential equations \[ \begin{aligned} \frac{d\omega_1}{d\tau} &=\omega_1\omega_2+\omega_1\omega_3-\omega_2\omega_3 \\ \frac{d\omega_2}{d\tau} &=\omega_1\omega_2+\omega_2\omega_3-\omega_1\omega_3 \\ \frac{d\omega_3}{d\tau} &=\omega_1\omega_3+\omega_2\omega_3-\omega_1\omega_2 \end{aligned} \] are \[ \omega_1=\frac{1}{2}\frac{d\log\theta_2^4(0,\tau)}{d\tau},\qquad \omega_2=\frac{1}{2}\frac{d\log\theta_3^4(0,\tau)}{d\tau},\qquad \omega_3= \frac{1}{2}\frac{d\log\theta_4^4(0,\tau)}{d\tau}, \] where \(\theta_i\) are the classical theta functions. In this paper, the author establishes a similar result for the following system of equations: \[ \begin{aligned} \frac{dX_\infty}{d\tau}&=\sum_{k=0}^{10} X_\infty X_k -X_\infty X_0-X_1 X_2 - X_3X_4-X_5X_6-X_7X_8-X_9X_{10},\\ \frac{dX_j}{d\tau} &= X_\infty X_j+\sum_{k=0}^{10} X_jX_k-X_j^2 -X_\infty X_0 -X_1X_2-X_3X_4-X_5 X_6-X_7X_8-X_9X_{10},\end{aligned} \] for \(0\leq j\leq 10\), where \(X_\infty\) and \(X_j\)'s satisfy certain relations. The author explicitly determines these \(X_j\)'s, namely, \[ X_\infty =\frac{d\log{\alpha_1}}{d\tau}, \quad X_0 =\frac{d\log{\alpha_2}}{d\tau},\quad X_{2k+1}= \frac{d\log\left(\alpha_2+\frac{1}{2}(1-\sqrt{5})\varepsilon^k\alpha_1\right)} {d\tau}, \] and \[ X_{2k+2}= \frac{d\log\left(\alpha_2+\frac{1}{2}(1+\sqrt{5})\varepsilon^k\alpha_1\right)} {d\tau}, \] where \(0\leq k\leq 4\), \(\varepsilon=e^{2\pi i/5}\), \(q=e^{\pi i\tau}\), \[ \begin{aligned} \alpha_1 &= \prod_{n=1}^\infty(1-q^{2n})^{-3/5}\sum_{n\in \mathbb Z}(-1)^nq^{5n^2-n},\\ \text{and} \alpha_2 &= q^{2/5}\prod_{n=1}^\infty(1-q^{2n})^{-3/5}\sum_{n\in \mathbb Z}(-1)^nq^{5n^2-3n}. \end{aligned} \]
scientific article; zbMATH DE number 1910462