Super Automorphic Forms on the Super Upper Half Plane

TL;DR

This paper extends the theory of automorphic forms to the super upper half plane, incorporating odd directions and super Lie group actions, and analyzes their dimensions and stability under deformations.

Contribution

It introduces super automorphic and cusp forms on the super upper half plane, providing dimension formulas and demonstrating stability under local super deformations.

Findings

Derived asymptotic dimension formulas for super automorphic forms.

Showed stability of these forms under local super deformations.

Extended classical automorphic form theory to supergeometry context.

Abstract

The super upper half plane, this is the ordinary upper half plane with additional odd (anticommuting) directions, admits a transitive super action of a certain super Lie group G . First we define the spaces of super automorphic and cusp forms on the super upper half plane for an ordinary lattice in G and give an asymptotic formula for their dimensions for high weight. For involving also 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.