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3 Edited for consistency with the proposer's normalization (and of the definition of eta)

[Edited for consistency with the normalization $\theta(z) = \sum_{n=-\infty}^\infty e^{\pi i n^2 z}$ of the proposer (and of my own article!); expansions in powers of $e^{2\pi i z}$ are more common, but for this modular form $q = e^{\pi i z}$ does seem to be the better choice.]

The function $$\theta(z) = \sum_{n=-\infty}^\infty q^{n^2}$$ (where $q = e^{2\pi e^{\pi i z}$) has a product formula converging in $|q|<1$, and thus for all $z$ in the upper half-plane: $$\theta(z) = \frac{\eta(2z)^5}{\left(\eta(q) \eta(q^4)\right)^2} frac{\eta(z)^5}{\bigl(\eta(z/2)\eta(2z)\bigr)^2} = \frac{1+q}{1-q} \cdot \frac{1-q^2}{1+q^2} \cdot \frac{1+q^3}{1-q^3} \cdot \frac{1-q^4}{1+q^4} \cdot \frac{1+q^5}{1-q^5} \cdot \frac{1-q^6}{1+q^6} \cdots$$ (where $\eta(z) = q^{1/24e^{\pi i z/12} \prod_{m=1}^\infty (1-q^m)$ 1-e^{2\pi i m z})$as usual). It follows from this formula that$\theta$cannot vanish except at a cusp: as long as$|q|<1$, the factors converge quickly to$1$and none of them has a zero or pole. But$\theta$is a modular form of positive weight, so it must vanish somewhere. It is clear from the sum formula that$\theta \neq 0$as$q \rightarrow 1$and$q \rightarrow 0$, so$\theta$does not vanish at the cusps$z=0$and$z=\infty$. Hence it must vanish at the remaining cusp . of$\Gamma(2)$. Indeed each factor$(1 \pm q^n) / (1 \mp q^n)$vanishes at$q = -1$(and stays in$(0,1)$for$-1 < q < 0$), so$\theta$must vanish at the corresponding cusp$z=1/2$.z=1$.

The author of the question also asked for a source for "a good reference for this problem". It's not clear what level of exposition is most appropriate here. Perhaps this paper might be of use, since I needed to use the vanishing order of $\theta$ in a context where I could not assume the reader had any background in modular forms:

Elkies N.D.: A characterization of the ${\bf Z}^n$ lattice, Math. Research Letters 2 (1995), 321-326 (arxiv:math.NT/9906019).

2 inserted a couple of explanatory phrases about the behavior of the infinite products

The function $$\theta(z) = \sum_{n=-\infty}^\infty q^{n^2}$$ (where $q = e^{2\pi i z}$) has a product formula converging in $|q|<1$, and thus for all $z$ in the upper half-plane: $$\theta(z) = \frac{\eta(2z)^5}{\left(\eta(q) \eta(q^4)\right)^2} = \frac{1+q}{1-q} \cdot \frac{1-q^2}{1+q^2} \cdot \frac{1+q^3}{1-q^3} \cdot \frac{1-q^4}{1+q^4} \cdot \frac{1+q^5}{1-q^5} \cdot \frac{1-q^6}{1+q^6} \cdots$$ (where $\eta(z) = q^{1/24} \prod_{n=1}^\infty prod_{m=1}^\infty (1-q^n)$ 1-q^m)$as usual). It follows from this formula that$\theta$cannot vanish except at a cusp: as long as$|q|<1$, the factors converge quickly to$1$and none of them has a zero or pole. But$\theta$is a modular form of positive weight, so it must vanish somewhere. It is clear from the sum formula that$\theta \neq 0$as$q \rightarrow 1$and$q \rightarrow 0$, so$\theta$does not vanish at the cusps$z=0$and$z=\infty$. Hence it must vanish at the remaining cusp. Indeed each factor$(1 \pm q^n) / (1 \mp q^n)$vanishes at$q = -1$, 1$ (and stays in $(0,1)$ for $-1 < q < 0$), so $\theta$ must vanish at the corresponding cusp $z=1/2$.

The author of the question also asked for a source for "a good reference for this problem". It's not clear what level of exposition is most appropriate here. Perhaps this paper might be of use, since I needed to use the vanishing order of $\theta$ in a context where I could not assume the reader had any background in modular forms:

Elkies N.D.: A characterization of the ${\bf Z}^n$ lattice, Math. Research Letters 2 (1995), 321-326 (arxiv:math.NT/9906019).

1

The function $$\theta(z) = \sum_{n=-\infty}^\infty q^{n^2}$$ (where $q = e^{2\pi i z}$) has a product formula converging in $|q|<1$, and thus for all $z$ in the upper half-plane: $$\theta(z) = \frac{\eta(2z)^5}{\left(\eta(q) \eta(q^4)\right)^2} = \frac{1+q}{1-q} \cdot \frac{1-q^2}{1+q^2} \cdot \frac{1+q^3}{1-q^3} \cdot \frac{1-q^4}{1+q^4} \cdot \frac{1+q^5}{1-q^5} \cdot \frac{1-q^6}{1+q^6} \cdots$$ (where $\eta(z) = q^{1/24} \prod_{n=1}^\infty (1-q^n)$ as usual). It follows from this formula that $\theta$ cannot vanish except at a cusp. But $\theta$ is a modular form of positive weight, so it must vanish somewhere. It is clear from the sum formula that $\theta \neq 0$ as $q \rightarrow 1$ and $q \rightarrow 0$, so $\theta$ does not vanish at the cusps $z=0$ and $z=\infty$. Hence it must vanish at the remaining cusp. Indeed each factor $(1 \pm q^n) / (1 \mp q^n)$ vanishes at $q = -1$, so $\theta$ must vanish at the corresponding cusp $z=1/2$.

The author of the question also asked for a source for "a good reference for this problem". It's not clear what level of exposition is most appropriate here. Perhaps this paper might be of use, since I needed to use the vanishing order of $\theta$ in a context where I could not assume the reader had any background in modular forms:

Elkies N.D.: A characterization of the ${\bf Z}^n$ lattice, Math. Research Letters 2 (1995), 321-326 (arxiv:math.NT/9906019).