Selberg super-trace formula for super Riemann surfaces II: Elliptic and parabolic conjugacy classes, and Selberg super-zeta functions

Abstract

Further contributions developing a super analogue of the classical Selberg trace formula, the Selberg super-trace formula, are presented. This paper deals with the calculation of contributions arising from elliptic and parabolic conjugacy classes to the Selberg super-trace formula for super Riemann surfaces. Analytic properties and the functional equation for the corresponding Selberg super-zeta functionR 0,R 1 andZ S , respectively, are derived and discussed. In particular, the elliptic contributions to a super Fuchsian group only alter the multiplicities of the “trivial” zeros and poles of the Selberg super-zeta functionR 0,R 1 andZ S , respectively, already due to the hyperbolic conjugacy classes. The parabolic conjugacy classes introduce new features in the analytical structure.