Non-Archimedean L-Functions: Of Siegel and Hilbert Modular Forms (Lecture Notes in Mathematics)

1) p n=1 The set of arguments s for which ((s) is defined can be extended to all s E C,s :f:. 1, and we may regard C as the group of all continuous quasicharacters C = Hom(R~, c> This book is devoted to the arithmetical theory of Siegel modular forms and their L-functions. The central object are L-functions of classical Siegel modular forms whose special values are studied using the Rankin-Selberg method and the action of certain differential operators on modular forms which have nice arithmetical properties. A new method of p-adic interpolation of these critical values is presented. An important class of p-adic L-functions treated in the present book are p-adic L-functions of Siegel modular forms having logarithmic growth (which were first introduced by Amice, Velu and Vishik in the elliptic modular case when they come from a good supersingular reduction of ellptic curves and abelian varieties). The given construction of these p-adic L-functions uses precise algebraic properties of the arihmetical Shimura differential operator. The book could be very useful for postgraduate students and for non-experts giving a quick access to a rapidly developping domain of algebraic number theory: the arithmetical theory of L-functions and modular forms Front Matter....Pages N2-vii Introduction....Pages 1-8 Acknowledgement....Pages 8-8 Non-Archimedean analytic functions, measures and distributions....Pages 9-34 Siegel modular forms and the holomorphic projection operator....Pages 35-80 Non-Archimedean standard zeta functions of Siegel modular forms....Pages 81-116 Non-Archimedean convolutions of Hilbert modular forms....Pages 117-145 Back Matter....Pages 146-161 Addressing specialists in the fields of representation theory, functional analysis and algebraic geometry, this monograph stresses the arithmetic of zeta functions of automorphic forms. It provides background on p-adic measures, their Mellin transforms and Euler products. Alexey A. Panchishkin. Subseries: Mathematisches Institut Der Universität Und Max-planck-institut Für Mathematik, Bonn -- Vol. 16. Includes Bibliographical References (p. [146]-154) And Index.