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

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?

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 $$1/2$$.