Modular Forms of Half Integral Weight

The forms to be discussed are those with the automorphic factor (cz + d)k/2 with a positive odd integer k. The theta function $$ \theta \left( z \right) = \sum\nolimits_{n = - \infty }^\infty {e^{2\pi in^2 z} } $$ and the Dedekind eta function $$ \eta \left( z \right) = e^{\pi iz/12} \prod _{n = 1}^\infty (1 - e^{2\pi inz} ) $$ are classical examples of such forms. (For some practical reasons, we take \( e^{2\pi in^2 z} \) instead of the usual \( e^{\pi in^2 z} \) in the definition of θ.) In fact, the function θ satisfies $$ \theta (\gamma (z)) = j(\gamma ,z)\theta (z)for all\gamma \in \Gamma _0 (4) $$ (1.1) with $$ j([\begin{array}{*{20}c} a & b \\ c & d \\ \end{array} ],z) = (\frac{c} {d})\varepsilon _d^{ - 1} (cz + d)^{1/2}. $$ (1.2) .