Conjectures in Arithmetic Algebraic Geometry A Survey
In this expository text we sketch some interrelations between several famous conjectures in number theory and algebraic geometry that have intrigued math ematicians for a long period of time. Starting from Fermat's Last Theorem one is naturally led to in
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Wilfred W. J. Hulsbergen
Conjectures in Arithmetic Algebraic Geometry A Survey Second Edition
Wilfred W. j. Hulsbergen Conjectures in Arithmetic Algebraic Geometry
Asped~f
Mathematic~
Edited by Klas Diederich Vol. E 2:
M. Knebusch/M. Kolster: Wittrings
Vol. E 3:
G. Hector/U. Hirsch: Introduction to the Geometry of Foliations, Part B
Vol. E 5:
P. Stiller: Automorphic Forms and the Picard Number of an Elliptic Surface
Vol. E 6:
G. Failings/G. Wustholz et al.: Rational Points*
Vol. E 7:
W. Stoll: Value Distribution Theory for Meromorphic Maps
Vol. E 9:
A. Howard/P.-M. Wong (Eds.): Contribution to Several Complex Variables
Vol. E 10: A. J. Tromba (Ed.): Seminar of New Results in Nonlinear Partial Differential Equations* Vol. E 13: Y. Andre: G-Functions and Geometry* Vol. E 14: U. Cegrell: Capacities in Complex Analysis Vol. E 15: J.-P. Serre: Lectures on the Mordeii-Weil Theorem Vol. E 16: K. lwasaki/H. Kimura/S. Shimomura/M. Yoshida: From Gauss to Painleve Vol. E 17: K. Diederich (Ed.): Complex Analysis Vol. E 18: W. W. J. Hulsbergen: Conjectures in Arithmetic Algebraic Geometry Vol. E 19: R. Rocke: Lectures on Nonlinear Evolution Equations Vol. E 20:
F. Hirzebruch, Th. Berger, R. Jung: Manifolds and Modular Forms*
Vol. E 21:
H. Fujimoto: Value Distribution Theory of the Gauss Map of Minimal Surfaces in Rm
Vol. E 22:
D.
Vol. E 23:
A. P. Fordy/J. C. Wood (Eds.): Harmonic Maps and Integrable Systems
Vol. E 24:
D. S. Alexander: A History of Complex Dynamics
V. Anosov/ A. A. Bolibruch: The Riemann-Hilbert Problem
*A Publication of the Max-Pianck-lnstitut fur Mathematik, Bonn Volumes of the German-language subseries ·Aspekte der Mathematik" are listed at the end af the book.
Conjectures in Arithmetic Algebraic Geollletry Wilfred W. J. Hulsbergen
A Survey Second Revised Edition
II Vleweg
Wilfred W. J. Hulsbergen KMA, NL-4800 RG Breda The N etherlands
AMS subject classification: llG, llM, 14C, 14G, 14H, 14K, 190, 19E, 19F.
First Edition 1992 Second Revised Edition 1994 Ali rights reserved © Springer Fachmedien Wiesbaden 1994 Originally published by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH. BraunschweiglWiesbaden in 1994
No part of this publication may be reproduced, stored in a retrieval system or transmitted, mechanical, photocopying or otherwise without prior permission of the copyright holder.
Cover design: Wolfgang Nieger, Wiesbaden Printed on acid-free paper
ISSN 0179-2156 ISBN 978-3-663-09507-1 ISBN 978-3-663-09505-7 (eBook) DOI 10.1007/978-3-663-09505-7
Contents Introduction
1
1 The 1.1 1.2 1.3 1.4 1.5
zero-dimensional case: number fields Class Numbers . . . . . . . Dirichlet L-Functions . . . . The Class Number Formula Abelian Number Fields . . . Non-abelian Number Fields and Artin L-Functions
5 5 8
2 The 2.1 2.2 2.3 2.4 2.5 2.6 2. 7 2.8
one-dimensional case: elliptic curves General Features of Elliptic Curves Varieties over Finite Fields . . . . . . . . . L-Functions of Elliptic Curves . . . . . . . Complex Multiplication and Modular Elliptic Curves Arithmetic of
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