Pseudodifferential Analysis, Automorphic Distributions in the Plane and Modular Forms

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 g-transforms, for g E SL(2;Z), of some initial function, say in S(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.

The book is addressed to a wide audience of advanced graduate students and researchers working in analytic number theory or pseudo-differential analysis.

From the reviews:

“The book is devoted to explaining a close relationship between the pseudodifferential analysis and the well-known theory of automorphic functions and modular forms on the upper Poincaré half-plane II, or their generalization as automorphic distributions. … the book is perfectly readable and rich with analytic details for both researchers in pseudodifferential analysis and for number theorists.” (Đȭ Ngọc Diệp, Mathematical Reviews, November, 2013)

“In this book the author explains very beautiful links between pseudodifferential analysis and the theory of nonholomorphic modular forms on the classical modular group … . The book is excellently written and represents an extremely valuable contribution for the two research communities – analysts from PDEs and pseudodifferential operators and number theorists. It exhibits a lot of new and original links between the two research areas. It is self-contained and easily accessible for a broad readership.” (Sören Kraußhar, Zentralblatt MATH, Vol. 1243, 2012)