What are the modularity properties of Weierstrass sigma function?

Question

I'm a little confused at the sigma orientation of tmf, see e.g. Witten genus and its references. The Weierstrass sigma function can be written as $$\sigma_L(z)(q)=\frac{z}{\exp\left(\sum_{k\ge 2} G_k(q) \frac{z^k}{k!}\right)}$$ and is the exponential for the sigma orientation of Tate K-theory over $\mathbb{Q}[[q]]$. Doesn't the sigma orientation correspond to the universal elliptic formal group law in the neighborhood of that cusp? And as such, shouldn't $\sigma_L^{-1}(z)(q)$ be a logarithm for the universal elliptic formal group law?

If so, its inverse (in $z$) should be some kind of weight-1 modular form in $q$, no? A logarithm for the universal formal group law has the same coefficients as an invariant differential fiberwise, so it should transform like a section of the pushforward of the relative differentials $\pi_* \Omega_{E/S}$, whose global sections down below are weight-1 modular forms.

However, the expression I wrote above doesn't really seem to transform in the correct way; I've tried computing the action of $\Gamma$ on $\sigma_L^{-1}$ with the usual substitution $\tau\mapsto \frac{a\tau + b}{c\tau + d}$ and $z\mapsto \frac{z}{c\tau + d}$, but all I end up getting is a mess. Is this coordinate $z$ not the usual coordinate coming from $(\tau, z)\in \mathbb{H} \times \mathbb{C}$?

Where am I going wrong?

Answer 1

The classical Weierstrass sigma function is not exactly $\sigma_L$ but rather, with your notations, $\sigma(z,\tau)=e^{az^2} \sigma_L(2\pi iz)(q)$ for some constant $a$, see Remark 5.3 in Ando, Basterra, The Witten genus and equivariant elliptic cohomology. (It is more standard to put $u=e^{2\pi iz}$ instead of $u=e^z$, this is for example the convention in Silverman's books.)

The modularity property cannot be found using the infinite $q$-product of $\sigma$. Rather we can use the definition of $\sigma$, namely a product over the lattice $\mathbb{Z}+\tau\mathbb{Z}$. The link between the two representations is worked out in details in Silverman, Advanced topics in the arithmetic of elliptic curves (Chapter 1, Sections 5 and 6). You're right that $\sigma$ basically transforms like a modular form of weight -1. Using the product over the lattice it is immediate that \begin{equation*} \sigma\bigl(\frac{z}{c\tau+d},\frac{a\tau+b}{c\tau+d}\bigr) = (c\tau+d)^{-1} \sigma(z,\tau) \end{equation*} for any $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ in $\mathrm{SL}_2(\mathbb{Z})$.

The way to produce modular forms out of that is to specialize $z=\alpha+\beta\tau$ where $\alpha,\beta$ are fixed rational numbers (the level of the modular form is the denominator of $(\alpha,\beta)$). But here it won't work as easily because $\sigma$ is not periodic with respect to $z$, you have to take suitable products or quotients of specializations to get true modular forms.

EDIT. Actually we can also use the formula you mention to show the modularity property. The only nontrivial step is the transformation rule for the Eisenstein series $G_2$, which is only quasimodular. This is classical and can be found for example in the book of Cohen and Strömberg on modular forms. The quasi-modularity will then correspond to this additional factor $e^{az^2}$.