Arithmetical Investigations Representation Theory, Orthogonal Polyno
In this volume the author further develops his philosophy of quantum interpolation between the real numbers and the p-adic numbers. The p-adic numbers contain the p-adic integers Zp which are the inverse limit of the finite rings Z/pn. This gives ris
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Shai M.J. Haran
Arithmetical Investigations Representation Theory, Orthogonal Polynomials, and Quantum Interpolations
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Shai M.J. Haran Department of Mathematics Technion – Israel Institute of Technology Haifa, 32000 Israel [email protected]
ISBN 978-3-540-78378-7 e-ISBN 978-3-540-78379-4 DOI: 10.1007/978-3-540-78379-4 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008921367 Mathematics Subject Classification (2000): 11-02, 11S80, 11S85 c 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH Printed on acid-free paper 987654321 springer.com
To Yedidya, Antonia, Elisha, Yehonadav, Amiad & Yoad.
Preface
This book grew out of lectures given at Kyushu University under the support of the Twenty-first Century COE Program “Development of Dynamical Mathematics with High Functionality” (Program Leader: Prof. Mitsuhiro Nakao). They were meant to serve as a primer to my book [Har5]. Indeed that book is very condense, and hard to read. We included however many new themes, such as the higher rank generalization of [Har5], and the fundamental semigroup. Since the audience consisted mainly of representation theorists, the focus shifted more into representation theory (hence less into geometry). We kept the lecture flair, sometimes explaining basic material in more detail, and sometimes only giving brief descriptions. This book would have never come to life without the many efforts of Professor Masato Wakayama. The author thanks him also for his incredible hospitality. Thanks are also due to Yoshinori Yamasaki, who did an excellent job of writing down and typing the lectures into LATEX. July 2006
Haifa
Contents
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Introduction: Motivations from Geometry . . . . . . . . . . . . . . . . . 1 0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.2 Analogies Between Arithmetic and Geometry . . . . . . . . . . . . . . . 2 0.3 Zeta Function for Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.4 The Riemann–Roch Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 0.5 The Castelnuovo–Sev
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