2
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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 $1/2$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.