Pseudodifferential Analysis, Automorphic Distributions in the Plane and Modular Forms

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<p>Pseudodifferential analysis, introduced in this book in a way adapted to the needs of number theorists, relates automorphic function theory in the hyperbolic half-plane Π to automorphic distribution theory in the plane. Spectral-theoretic questions are discussed in one or the other environment: in the latter one, the problem of decomposing automorphic functions in Π according to the spectral decomposition of the modular Laplacian gives way to the simpler one of decomposing automorphic distributions in R² into homogeneous components. The Poincaré summation process, which consists in building automorphic distributions as series of <i>g</i>-transforms, for <i>g E SL</i>(2<i>;</i>Z), of some initial function, say in <i>S</i>(R²), is analyzed in detail. On Π, a large class of new automorphic functions or measures is built in the same way: one of its features lies in an interpretation, as a spectral density, of the restriction of the zeta function to any line within the critical strip.</p><p><p>The book is addressed to a wide audience of advanced graduate students and researchers working in analytic number theory or pseudo-differential analysis.</p>