Transformation laws for theta functions (Q2782383)

Let \(Q\) be a positive-definite, integral quadratic form of rank \(2r\) and level \(N\). For a vector \(v\in \mathbb{C}^{2r}\), the authors define NEWLINE\[NEWLINE\Theta(Q,v,\tau,X)= \sum_{n=0}^\infty \sum_{m\in \mathbb{Z}^{2r}} \frac{2^n\langle v,m\rangle^{2n} e^{2\pi iQ(m)\tau}} {(2n)!} (2\pi iX)^nNEWLINE\]NEWLINE and set \(\varepsilon(k)= (-1)^r \frac{\det A}{k}\) for a positive integer \(k\), where \(A\) is a Gram matrix of the bilinear form \(\langle \cdot,\cdot\rangle\) corresponding to \(Q\). The main result of this article is as follows: for \(\gamma= \left(\begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \right)\in \Gamma_0(N)\), NEWLINE\[NEWLINE\Theta \biggl(Q,v, \gamma(\tau), \frac{X}{(c\tau+d)^2} \biggr)= \varepsilon(d) (c\tau+d)^r \exp\biggl( \frac{c\langle v,v\rangle X}{c\tau+d} \biggr) \Theta(Q,v,\tau,X).NEWLINE\]NEWLINENEWLINENEWLINEFor the entire collection see [Zbl 0980.00029].