Bergman space zero sets, modular forms, von Neumann algebras and ordered groups

Abstract

$A^2_{\alpha}$ will denote the weighted $L^2$ Bergman space. Given a subset $S$ of the open unit disc we define $\Omega(S)$ to be the infimum of $\{s| \exists f \in A^2_{s-2}, f\neq 0, \mbox{ having $S$ as its zero set} \}$.By classical results on Hardy space there are sets $S$ for which $\Omega(S)=1$. Using von Neumann dimension techniques and cusp forms we give examples of $S$ where $1<\Omega(S)<\infty$. By using a left order on certain Fuchsian groups we are able to calculate $\Omega(S)$ exactly if $\Omega (S)$ is the orbit of a Fuchsian group. This technique also allows us to derive in a new way well known results on zeros of cusp forms and indeed calculate the whole algebra of modular forms for \pslz.