Index and weight of Weierstrass $\sigma$-function as a Jacobi form, versus a statement in a note by Zagier

Question

Let $$\sigma_L(w; \tau):=\frac{w}{\exp\left(\sum_{k\ge 2} 2G_k(q)\frac{w^k}{k!}\right)}$$ be the version of the Weierstrass $\sigma$-function which is used to orient $\text{tmf}$; here $w=2\pi i z$, $q=\exp(2\pi i \tau)$, $G_k=\frac{(k-1)!}{2(2\pi i)^k}E_k$ (nonzero only for even $k$) is the modular Eisenstein series normalized to have constant Fourier coefficient $-B_k/2k$ for $k\ge 4$, and $G_2$ analogously is the quasimodular weight-$2$ Eisenstein series which transforms as $$G_2\left(\frac{a\tau+b}{c\tau + d}\right)=(c\tau+d)^2G_2(\tau)+\frac{ic(c\tau+d)}{4\pi}$$ under the action of $\text{SL}_2(\mathbb{Z})$.

I'm somewhat confused about this seemingly straightforward issue: what is the weight and index of $\sigma_L$ as a Jacobi form?

Recall that a Jacobi form of weight $k$ and index $m$ for $\Gamma = \text{SL}_2(\mathbb{Z})$ is a holomorphic function $(\tau, z)\mapsto \phi(\tau, z)$ on $\mathcal{H}\times \mathbb{C}$ satisfying a modular transformation property

$$\phi\left(\frac{a\tau + b}{c\tau +d}, \frac{z}{c\tau+d}\right)= (c\tau+d)^k \exp(2\pi i m c z^2/(c\tau +d))\phi(\tau, z)$$

and also an elliptic transformation property I won't mention since I won't be using it here. It's my understanding that $\sigma_L(2\pi i z; \tau)$ should be a Jacobi form, and indeed we can check the modular transformation law:

$$\sigma_L\left(\frac{w}{c\tau+d},\frac{a\tau + b}{c\tau +d}\right)=\frac{1}{c\tau + d}\frac{w}{\exp\left(\left[(c\tau + d)^22G_2(q) + 2ic(c\tau+d)/4\pi]\frac{w^2}{2(c\tau + d)^2}+\sum_{k\ge 4} 2(c\tau + d)^kG_k(q)\frac{w^k}{k!(c\tau + d)^k}\right)\right]}$$

$$ = \frac{1}{c\tau + d}\frac{w}{\exp\left(\sum_{k\ge 2} 2G_k(q) \frac{w^k}{k!}\right)}\exp\left(\frac{-icw^2}{4\pi (c\tau+d)}\right)$$

$$ = \frac{1}{c\tau + d}\exp\left(\frac{-ic(2\pi iz)^2}{4\pi (c\tau+d)}\right)\sigma_L(w,\tau)=\frac{1}{c\tau + d}\exp\left(\frac{\pi icz^2}{c\tau+d}\right)\sigma_L(w,\tau).$$

seemingly giving it as weight $-1$ and index $1/2$. However, on the last page of, Zagier's note on the Landweber-Stong genus, he claims that $u^{-1}P_W(u)$ in his notation (which is $\sigma_L^{-1}$ in ours) is weight $-1$ and index $-1/2$, which would imply $\sigma_L$ is weight $1$ and index $1/2$. What's going on here? Is this just a typo in that note?

Answer 1

It appears to be a typo. The equation just above that claim, $$u^{-1} P_W(u) = \Big( \sum_{n \in \mathbb{Z}} \left( \frac{-4}{n} \right) n q^{n^2 / 8} \Big) \Big/ \Big( \sum_{n \in \mathbb{Z}} \left( \frac{-4}{n} \right) q^{n^2 / 8} e^{nu / 2} \Big),$$ or in other notation, $$u^{-1} P_W(u) = \eta^3(\tau) / \vartheta_{11}(\tau; u)$$ immediately shows that $u^{-1} P_W(u)$ has weight $1$, rather than $-1$: because $\eta$ and $\vartheta_{11}$ both have weight $1/2$.