Period functions for Maass wave forms. I.

Lewis, J.; Zagier, D.

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From pseudodifferential analysis to modular form theory

André Unterberger (1999)

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Taking advantage of methods originating with quantization theory, we try to get some better hold - if not an actual construction - of Maass (non-holomorphic) cusp-forms. We work backwards, from the rather simple results to the mostly devious road used to prove them.

The cubic moment of central values of automorphic $L$ -functions.

Conrey, J.B., Iwaniec, H. (2000)

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An analytic function and iterated integrals

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A spectral analysis of automorphic distributions and Poisson summation formulas

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Automorphic distributions are distributions on $ℝ^{d}$ , invariant under the linear action of the group $S L (d, ℤ)$ . Combs are characterized by the additional requirement of being measures supported in $ℤ^{d}$ : their decomposition into homogeneous components involves the family ${(𝔈_{i λ}^{d})}_{λ \in ℝ}$ , of Eisenstein distributions, and the coefficients of the decomposition are given as Dirichlet series $𝒟 (s)$ . Functional equations of the usual (Hecke) kind relative to $𝒟 (s)$ turn out to be equivalent to the invariance of the comb under some modification...