Pseudodifferential Analysis, Automorphic Distributions in the Plane and Modular Forms

ISBN-13: 9783034801652 / Angielski / Miękka / 2011 / 300 str.

Andre Unterberger

Opis

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 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 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 Poincare 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, 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.

Recenzje

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." ( Ng c Di p, 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)

Spis treści

Introduction.- The Weyl calculus.- The Radon transformation and applications.- Automorphic functions and automorphic distributions.- A class of Poincaré series.- Spectral decomposition of the Poincaré summation process.- The totally radial Weyl calculus and arithmetic.- Should one generalize the Weyl calculus to an adelic setting?.- Index of notation.- Subject Index.- Bibliography.

Blurb

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 Î 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.