Meromorphic and rational factors of automorphy (Q1074624)

Let \(\nu\) (z,N) be a factor of automorphy on \({\mathbb{C}}\times \Gamma\) where \(\Gamma\) \(\subset SL(2,{\mathbb{R}})\), i.e. \(\nu (z,MN)=\nu (z,N) \nu (Nz,M)\) for all M,N\(\in \Gamma\). If all matrices of \(\Gamma\) have lower left entries zero and \(\Gamma\) is discrete then to any factor of automorphy \(\nu\) meromorphic on the complex plane there is an entire nonvanishing h(z) such that \(\nu (z,N)=w(z,N)h(Nz)/h(z)\) for all N in \(\Gamma\) with an auxiliary function w(z,N) defined in the paper. If now \(\Gamma\) contains a matrix \(\left( \begin{matrix} a\\ c\end{matrix} \begin{matrix} b\\ d\end{matrix} \right)\) with \(c\neq 0\) then the meromorphic factor of automorphy \(\nu\) (z,M) is rational in z for all M in \(\Gamma\). With the additional assumptions that the factor of automorphy is rational, i.e. that \(\nu\) (z,M) is proportional to \(\prod^{\alpha}_{i=1}(z-x_ i)/\prod^{\beta}_{i=1}(z-y_ i)\) where \(\alpha\), \(\beta\) do not exceed some fixed integer n, the technical assumptions that no \(x_ i\) or \(y_ i\) is the orbit of infinity, and that \(\Gamma\) contains either a matrix of infinite order or a matrix of order \(\geq 4n^ 4\), then there exist disjoint lists \(\{z_ i\}\) and \(\{\zeta_ i\}\), and fixed integer \(\alpha\) such that for every M in \(\Gamma\) then \(\nu\) (z,M) is proportional to \[ \prod^{\alpha}_{i=1}(z-z_ i)/(z-M^{-1}z_ i)\cdot \prod^{n-\alpha}(z-M^{-1}\zeta_ i)/\quad (z-\zeta_ i). \] The corresponding automorphic forms f(z) are also determined.