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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?

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  • $\begingroup$ Why do you think that the Jacobi theta functions are Jacobi forms? $\endgroup$
    – Somos
    Apr 18 '18 at 0:46
  • $\begingroup$ Well because they look schematically like that general form in the first equation of my OP in the case of N=1. And in fact, just after Eichler and Zagier write that formula they mention that it's a generalization of what Jacobi studied for N=1. I figured they were referring to these theta functions. $\endgroup$
    – Benighted
    Apr 18 '18 at 1:05
  • $\begingroup$ That is a good answer, but even for N=1, specializing a Jacobi form doesn't mean you can get exactly a Jacobi theta function. Some adjustments have to be made. Eichler and Zagier should have given explicit details of how, for example, $\vartheta_3$ is a specialization of a Jacobi form. Did they? $\endgroup$
    – Somos
    Apr 18 '18 at 1:50
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    $\begingroup$ Eichler and Zagier were certainly inspired by Jacobi theta functions, but the definition they give of Jacobi forms requires that the weight and index are integers. The Jacobi theta functions you give are weight one-half and so are not Jacobi forms by the definition given in Eichler-Zagier. $\endgroup$ Apr 18 '18 at 2:13
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    $\begingroup$ Theta functions with half integral weight are best thought of as vector-valued forms which transform under the Weil representation of the metaplectic group which is a double cover of $SL(2, \mathbb{Z})$. $\endgroup$ Apr 18 '18 at 14:08
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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 $$\vartheta(\tau,z+\lambda \tau + \mu) = (-1)^{\lambda +\mu} q^{-\lambda^2 / 2} \zeta^{-\lambda} \vartheta(\tau,z).$$

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