Selberg zeta functions associated with a theta multiplier system of \(\mathrm{SL}_2(\mathbb Z)\) and Jacobi forms
Certain Selberg zeta functions associated with a theta multiplier system of \(\mathrm{SL}_2(\mathbb Z)\) are defined. The Selberg zeta functions are analytically continued to meromorphic functions in the whole complex plane satisfying certain functional equations by virtue of the general theory of Selberg trace formulas and Selberg zeta functions. The Selberg trace formula discussed in this paper is closely related to the spaces of Jacobi forms. Denote by \(J_{k,m}\) (resp. \(J^*_{k,m})\) the space of holomorphic Jacobi forms (resp. skew-holomorphic Jacobi forms) of weight \(k\) and index \(m\) with respect to \(\Gamma=\mathrm{SL}_2(\mathbb Z)\). Certain relations between the dimensions of the spaces of \(J_{k,m}\), \(J^*_{k,m}\) \((k=1,2)\) and the orders of the zeros at \(s=3/4\) of the Selberg zeta functions are obtained.
scientific article; zbMATH DE number 4216037