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New models for the action of Hecke operators in spaces of Maass wave forms

Ian Kiming — 2007

Annales de l’institut Fourier

Utilizing the theory of the Poisson transform, we develop some new concrete models for the Hecke theory in a space M λ ( N ) of Maass forms with eigenvalue 1 / 4 - λ 2 on a congruence subgroup Γ 1 ( N ) . We introduce the field F λ = ( λ , n , n λ / 2 n ) so that F λ consists entirely of algebraic numbers if λ = 0 . The main result of the paper is the following. For a packet Φ = ( ν p p N ) of Hecke eigenvalues occurring in M λ ( N ) we then have that either every ν p is algebraic over F λ , or else Φ will – for some m – occur in the first cohomology of a certain space...

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