Weight, Index, and Congruence Subgroup of Classical Jacobi Theta Functions

Question

On the very first page in the Introduction of Eichler and Zagier's text on Jacobi forms, they mention that the theta function

$$\Theta_{x_{0}}(\tau, z) = \sum_{x \in \mathbb{Z}^{N}} q^{Q(x)} y^{B(x, x_{0})}$$

is a holomorphic Jacobi form of weight $N/2$ and index $Q(x_{0})$ for some congruence subgroup of $SL_{2}(\mathbb{Z})$. In this formula, $Q: \mathbb{Z}^{N} \to \mathbb{Z}$ is a positive-definite quadratic form, $B(x, x_{0}) = \frac{1}{2}(Q(x + x_{0}) - Q(x) - Q(x_{0}))$ is the associated bilinear form, and $x_{0}$ is some lattice vector. Finally, $q = e^{2 \pi i \tau}$ and $y = e^{2 \pi i z}$, and these are conventions I'd like to stay with for my purposes.

I would like to apply this to the example of the four classical Jacobi theta functions, but I haven't found any conclusive references, and those references I have found, all seem to use different conventions. I believe the Jacobi theta functions are given for my definition above of $q$ and $y$ by

$$\vartheta_{1}(\tau, z) = - \sum_{n \in \mathbb{Z}} q^{\frac{1}{2}(n + \frac{1}{2})^{2}}(-y)^{n+\frac{1}{2}}$$

$$\vartheta_{2}(\tau, z) = \sum_{n \in \mathbb{Z}} q^{\frac{1}{2}(n + \frac{1}{2})^{2}} y^{n + \frac{1}{2}}$$

$$\vartheta_{3}(\tau, z) = \sum_{n \in \mathbb{Z}} q^{n^{2}/2} y^{n}$$

$$\vartheta_{4}(\tau, z) = \sum_{n \in \mathbb{Z}} q^{n^{2}/2} (-y)^{n}$$

My questions simply is, what is the weight and index of each of these theta functions, and with respect to what congruence subgroup? Is there a nice reference?

I'm struggling to even reconcile these functions with the general theta series above from Eichler and Zagier. For example, $n^{2}/2$ is not an integer-valued quadratic form, and even if it were, $n$ has the wrong coefficient to be the corresponding bilinear form. This leads me to worry that I'm using the wrong conventions for $q$ and $y$. Moreover, don't the factors of $(-1)^{n+1/2}$ and $(-1)^{n}$ in $\vartheta_{1}$ and $\vartheta_{4}$ respectively, prevent us from putting these in the general form of $\Theta_{x_{0}}(\tau, z)$ above?

Answer 1

To expand on the comments. All four of the functions in the question are somehow versions of the classical theta function \begin{align*} \vartheta(\tau,z) &= \sum_{n \in \mathbb{Z}} (-1)^n q^{\frac{1}{2}(n+1/2)^2} \zeta^{n + 1/2} \\ &= \zeta^{1/2} q^{1/8} \prod_{n=1}^{\infty} (1 - q^n) (1 - q^n \zeta) (1 - q^{n-1} \zeta^{-1}), \end{align*} where $q = e^{2\pi i \tau}$ and $\zeta = e^{2\pi i z}$. This is indeed a Jacobi form, of half-integer weight and half-integer index and it is not covered explicitly by Eichler-Zagier. To make sense of it you should interpret it as having a multiplier system under $SL_2(\mathbb{Z})$ (or transforming under a character of the metaplectic group) as well as a character of the Heisenberg group.

If you let $\chi$ denote the character of Dedekind's eta function, so $$\eta((M,\sqrt{c \tau + d}) \cdot \tau) = \chi(M,\sqrt{c \tau + d}) \sqrt{c \tau + d} \eta(\tau)$$ then I believe the behavior of $\vartheta$ is $$\vartheta\Big( (M,\sqrt{c \tau + d}) \cdot \tau, \frac{z}{c \tau + d} \Big) = \chi(M,\sqrt{c \tau + d})^3 \sqrt{c \tau + d} e^{\pi i \frac{c z^2}{c \tau + d}} \vartheta(\tau,z)$$ and $$\vartheta(\tau,z+\lambda \tau + \mu) = (-1)^{\lambda +\mu} q^{-\lambda^2 / 2} \zeta^{-\lambda} \vartheta(\tau,z).$$