Reformulating Theta Function Symmetry as a Modular Form

Question

If $\theta$ is the Jacobi theta function $\theta(\tau) = \sum e^{\pi i n^2 \tau}$, then $\theta$ satisfies the Modular symmetries $\theta(\tau + 2) = \theta(\tau)$ and $\theta(-1/\tau) = \sqrt{-i \tau} \cdot \theta(\tau)$. Even if we square things, this isn't really completely the symmetry that a modular form should satisfy, i.e. $\theta^2(-1/z) = - i \tau \cdot \theta(\tau)$ whereas a modular form $f$ of weight one should satisfy $f(-1/\tau) = \tau f(\tau)$. Is there a standard way of working with the $\theta$ function so we can treat it, or powers of it, as actual modular forms?

Answer 1

The function $\theta^2(z)$ is a weight $1$ modular form on $\Gamma_0(4)$ with character $\chi_{-1}$. That is, it satisfies $$ \theta^2(\gamma z) = \left( \tfrac{-1}{d} \right) (cz + d) \theta^2(z), \qquad \gamma = \left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right).$$

This fits nicely in the general philosophy of modular forms with character or modular forms with nebentypus.

I should note that one can also study $\theta(z)$ as a half-integral weight modular form on a double-cover of $\Gamma_0(4)$.