I'm a little confused at the sigma orientation of tmf, see e.g. [Witten genus](https://ncatlab.org/nlab/show/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?



What are the modularity properties of Weierstrass sigma function?
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](https://arxiv.org/abs/math/0008192). (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 $e^{az^2}$.


