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May 8, 2020

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

$A^2_{\alpha}$ 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 | there exists f in A^2_{s-2}, f\neq 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.