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

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

2009-02-27

arxiv.org/abs/0902.4852v3

Mathematics

Complex Variables

48 pages, no figures. Replacement because of essential improvement of the result

Scientific paper

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.

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