In this article, the authors interpret a certain special function, the elliptic Gamma function, defined as
$$ \Gamma(z,\tau,\sigma)=\prod_{j,k=0}^\infty\frac{1-e^{2\pi i((j+1)\tau+(k+1)\sigma-z)}}{1-e^{2\pi i(j\tau+k\sigma+z)}} $$
with $z,\tau,\sigma\in\mathbb{C}$ and $\mathrm{Im}\,\tau,\mathrm{Im}\,\tau>0$, as the (values of a) generator of an "automorphic form of degree 1", obeying the equation
$$ \Gamma(z/\sigma,\tau/\sigma,-1/\sigma)=e^{i\pi Q(z;\tau,\sigma)}\Gamma((z-\sigma)/\tau,-1/\tau,-\sigma/\tau)\Gamma(z,\tau,\sigma), $$
for some polynomial $Q(z;\tau,\sigma)$, where the arguments of the Gamma functions are related by the action of some elements of $\mathrm{SL}\mathbb{(3,Z)}$. They call the Jacobi theta function an automorphic form of degree 0, and I am confused by this. I have two questions:
- What is the definition of the degree of an automorphic function in this context? I could not find anything in the literature that matches the authors' usage of the term.
- I thought that automorphic forms satisy equations of the form $f(g\cdot X)=j_g(X)f(X)$, $g$ being the element of some group $G$ acting on a complex manifold $X$, and $j_g(X)$ being the "factor of automorphy", a nowhere zero function. The above equation for the elliptic Gamma function is not of this form, containing three factors of the function instead of just two, and once again I could not find any other examples in the literature. How does the elliptic Gamma function match the definition of an automorphic form and are there other known examples of this type?
