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

TL;DR

This paper explores the zero sets of weighted Bergman spaces, employing von Neumann algebra techniques and modular forms to precisely determine the infimum of parameters for which these sets are zeros, revealing new insights into their structure.

Contribution

It introduces a novel method using von Neumann dimension and group orderings to compute zero set parameters and analyze modular form algebras.

Findings

Examples of zero sets with infimum between 1 and infinity.

Exact calculation of zero set parameters for Fuchsian group orbits.

New derivation of classical results on cusp form zeros.

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.