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The main results of this book combine pseudo differential analysis with modular form theory. The methods rely for the most part on explicit spectral theory and the extended use of special functions. The starting point is a notion of modular distribution in the plane, which will be new to most readers and relates under the Radon transformation to the classical one of modular form of the non-holomorphic type. Modular forms of the holomorphic type are addressed too in a more concise way, within a general scheme dealing with quantization theory and elementary, but novel, representation-theoretic concepts.

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Introduction.- Basic modular distributions.- From the plane to the half-plane.- A short introduction to the Weyl calculus.- Composition of joint eigenfunctions of (...) and (...).- The sharp composition of modular distributions.- The operator with symbol (...).- from non-holomorphic to holomorphic modular forms.- Index.

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"The book under review is devoted to developing a satisfactory Weyl calculus on the class of tempered automorphic distributions under the group SL(2, Z). ... The book is ... readable and is an interesting introduction to the subject. The main ideas are explained in detail and illustrated by exact computations." ( Ng c Di p, Mathematical Reviews, December, 2015)

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