Quotients of theta series as rational functions of \(j_{1,8}\) (Q2731055)

Let \(A\) and \(B\) be even unimodular definite quadratic forms in dimension \(n\geq 24\). The main result is the following: NEWLINE\[NEWLINE\frac{\theta_A(z)} {\theta_B(z)}= \frac{p(j_{1,8}^2(z))} {q(j_{1,8}^2(z))},NEWLINE\]NEWLINE where \(p,q\) are polynomials over \(\mathbb{Q}\) in \(j_{1,8}^2\) of degree \(\frac 12 (n-u\pmod{24})\) with \(j_{1,8}= \frac{\theta_3(2z)} {\theta_3(4z)}\) (with \(\theta_3\) the Jacobi theta series).