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

Abstract

$A_{\alpha }^2$ 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 $\{s\mid \exists f\in A_{s-2}^2,f\ne 0,$ 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 ${\mathrm{PLS}}_2(\mathbb{Z})$.