Non-Archimedean \(L\)-functions and arithmetical Siegel modular forms
The book is an updated version of the book `` Non-Archimedean \(L\)-Functions of Hilbert and Siegel Modular Forms'' by \textit{Alexei Panchishkin} published in 1991 (reviewed in Zbl 0732.11026), and uses the results of the Ph.D. Thesis of Michel Courtieu. The main subject of the book is the \(p\)-adic theory of \(L\)-functions of Siegel modular forms. The authors construct a \(p\)-adic analytic continuation of the standard \(L\)-functions of Siegel modular forms in a general case, using the techniques of the \(p\)-adic integration and the Rankin-Selberg convolution method. The basic new feature of this second version is the use of arithmetical nearly holomorphic Siegel modular forms (viewed as certain formal expansions of many variables with algebraic coefficients), and of arithmetical differential operators acting on nearly holomorphic Siegel modular forms. The book begins with a general introduction to the ideas of \(p\)-adic \(L\)-functions. Chapter 1 is an exposition of some standard facts from the theory of continuous and analytic functions over a non-Archimedean local field, admissible measures and their Mellin transforms, complex-valued distributions associated with Euler products, etc. Chapter 2 contains some preparatory facts concerning Fourier coefficients of the Siegel-Eisenstein series, holomorphic projection operator and the Maass operator. The purpose of Chapter 3 is to describe the action of certain arithmetical differential operators on algebraically defined nearly holomorphic Siegel modular forms. Finally, Chapter 4 is dedicated to give a new conceptual construction of admissible measures attached to a standard zeta function of a Siegel cusp form. For this purpose the authors use the theory of \(p\)-adic integration in spaces of nearly holomorphic Siegel modular forms. The book will be very useful for postgraduate students and researchers entering this difficult area of research.
scientific article; zbMATH DE number 2054649