Super automorphic forms on the super upper half plane

Abstract

Let H |r denote the upper half plane H with r additional odd (anticommuting) coordinates. It admits a transitive super action of a certain super Lie group 𝒢. First, we define the spaces of super automorphic and cusp forms on H |r for an ordinary lattice Γ of 𝒢, give an asymptotic formula for their dimensions for high weight and show how to embed Γ \ H |r into the super projective space with the help of super automorphic forms. For also involving the odd directions of 𝒢 we introduce local super deformation of lattices in 𝒢 and show that for high weight the spaces of super automorphic and cusp forms are stable under such local super deformations.