Non-Archimedean L-Functions and Arithmetical Siegel Modular Forms Se
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 cer
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Michel Courtieu Alexei Panchishkin
Non-Archimedean L-Functions and Arithmetical Siegel Modular Forms Second, Augmented Edition
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Authors Michel Courtieu Lyc´ee Anna de Noailles 2, Avenue Anna de Noailles 74500 Evian les Bains, France e-mail: [email protected]
Alexei A. Panchishkin Institut Fourier Universit´e Grenoble I BP 74 38402 Saint-Martin d’H`eres, France e-mail: [email protected]
Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de
Mathematics Subject Classification (2000): primary: 11F, 11R, 11S secondary: 19K, 46F, 46G ISSN 0075-8434 ISBN 3-540-40729-4 Springer-Verlag Berlin Heidelberg New York ISBN 3-540-54137-3 1st edition Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 1991, 2004 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10951675
41/3142/du - 543210 - Printed on acid-free paper
Preface
The present book is a updated version of the LNM 1471 "Non-Archimedean L-Functions of Hilbert and Siegel Modular Forms" by Alexei Panchishkin, appeared in 1991. A part of this new book uses the results of the PhD Thesis of Michel Courtieu (Grenoble, Institut Fourier, 2000). The main subject of the book is the p-aciic theory of L-functions of Siegel modular forms. In the case of the Riemann zeta functionand of the Dirichlet £-functions, this theory goes back to the classical Kummer congruences for Bernoulli numbers, and to their p-adic interpretation given by Kubota and Leopoldt, and by Mazur. Using the techniques of the p-adic integration, and of the Rankin-Selberg convolution method, we construct a p-adic analytic continuation of the standard L-functions of Siegel modular forms in a very general case, that is, for any non-zero Satake p-parameter. This second version has three basic new features, found re
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