Stability of the space of Automorphic Forms under Local Deformations of the Lattice

TL;DR

This paper studies how spaces of automorphic and cusp forms on the upper half-plane behave under local deformations of lattices in SL(2,R), showing they remain structurally stable and form free modules over the deformation parameter algebra.

Contribution

It introduces a framework for automorphic forms on P-lattices, demonstrating their stability and freeness as modules under local lattice deformations.

Findings

Spaces are free modules over the complexified P of rank equal to the original lattice's automorphic forms.

Automorphic forms adapt to local deformations, maintaining their structure.

The quotient space by the P-lattice forms a P-Riemann surface, enabling geometric interpretation.

Abstract

First we explain the concept of local deformation over a 'parameter' algebra P, in particular the notion of a P-lattice in a Lie group. Purpose of this article is to define the spaces of automorphic resp. cusp forms on the upper half plane H for a P- (!) lattice of SL(2, R) and to investigate their structure. It turns out that in almost all cases these spaces are free modules over the complexified P of rank equal to the dimension of the spaces of automorphic resp. cusp forms for the body, which is the associated ordinary lattice in SL(2, R) . In other words almost every automorphic resp. cusp form admits an 'adaption' to local deformations of the lattice. This is shown by giving the quotient of H by the P-lattice together with the cusps the structure of a P- Riemann surface and writing the spaces of automorphic resp. cusp forms as global sections of holomorphic P- (!) line bundles on this quotient.