© 1991

Non-Archimedean L-Functions of Siegel and Hilbert Modular Forms


Part of the Lecture Notes in Mathematics book series (LNM, volume 1471)

Table of contents

  1. Front Matter
    Pages N2-vii
  2. Alexey A. Panchishkin
    Pages 1-8
  3. Alexey A. Panchishkin
    Pages 8-8
  4. Alexey A. Panchishkin
    Pages 117-145
  5. Back Matter
    Pages 146-161

About this book


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.


11F 11R 11S 19K 46F 46G Eisenstein distributions measures modular forms analytic function convolution distribution zeta function

Authors and affiliations

  1. 1.Department of MathematicsMoscow State UniversityMoscowUSSR

Bibliographic information

  • Book Title Non-Archimedean L-Functions of Siegel and Hilbert Modular Forms
  • Authors Alexei A. Panchishkin
  • Series Title Lecture Notes in Mathematics
  • DOI
  • Copyright Information Springer-Verlag Berlin Heidelberg 1991
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Softcover ISBN 978-3-540-54137-0
  • eBook ISBN 978-3-662-21541-8
  • Series ISSN 0075-8434
  • Series E-ISSN 1617-9692
  • Edition Number 1
  • Number of Pages VII, 161
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Additional Information Originally published under: Panchishkin, A.A.
  • Topics Number Theory
    Algebraic Geometry
  • Buy this book on publisher's site
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From the reviews of the second edition:

"The book is an updated version of the book ‘Non-Archimedean L-Functions of Hilbert and Siegel Modular Forms’ by Alexei Panchishkin published in 1991 … . The main subject of the book is the p-adic theory of L-functions of Siegel modular forms. … The basic new feature of this second version is the use of arithmetical nearly holomorphic Siegel modular forms … . The book will be very useful for postgraduate students and researchers entering this difficult area of research." (Andrzej Dabrowski, Zentralblatt MATH, Vol. 1070, 2005)