Conjectures in Arithmetic Algebraic Geometry A Survey
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Coniectures in Arithmetic
Algebraic Geometry
As~
tvbthematics
Edited by Klas Diederich Vol. E1:
G. Hector/U. Hirsch: Introduction to the Geometry of Foliations, Part A
Vol. E2:
M. Knebusch/M. Kolster: Wittrings
VoI.E3:
G. Hector/U. Hirsch: Introduction to the Geometry of Foliations, Part B
Vol. E4:
M. Laska: Elliptic Curves of Number Fields with Prescribed Reduction Type lout of printl
Vol. E5:
P. Stiller: Automorphic Forms and the Picard Number of an Elliptic Surface
VoI.E6:
G. Faltings/G. WOstholz et 01.: Rational Points*
Vol. E7:
W Stoll: Value Distribution Theory for Meromorphic Maps
Vol. E8:
W. von Wahl: The Equations of Navier-Stokes and Abstract Parabolic Equations lout of printl
Vol. E9:
A. Howard/P.-M. Wong IEds.l: Contributions to Several Complex Variables
Vol. E 10:
A. J. Tromba IEd.l: Seminar of New Results in Nonlinear Partial Differential Equations*
Vol. Ell:
M. Yoshida: Fuchsian Differential Equations*
Vol. E 12:
R. Kulkarni, U. PinkaIiIEds.l: Conformal Geometry*
Vol. E13:
Y. Andre: G-Functions and Geometry*
Vol. E14:
U. Cegrell: Capacities in Complex Analysis
Vol. E15:
J.-P. Serre: Lectures on the Mordell-Weil Theorem
Vol. E 16:
K.lwasaki/H. Kimura/S. Shimomura/M. Yoshida: From Gauss to Painleve
Vol. E 17:
K. Diederich IEd.l: Complex Analysis
Vol. E18:
W. W. J. Hulsbergen: Conjectures in Arithmetic Algebraic Geometry
*A Publication of the Max-Planck-Institut fOr Mathematik, Bonn
Volumes of the German-language subseries "Aspekte der Mathematik" are listed at the end of the book.
Wilfred W J. Hulsbergen
Coniectures in ----..... Algebraic ASurvey
Wilfred W. J. Hulsbergen KMA, NL-4800 RG Breda The Netherlands
Die Deutsche Bibliothek - CIP-Einheitsaufnahme Hulsbergen, Wilfred W. J.: Conjectures in arithmetic algebraic geometry: a survey / Wilfred W. J. Hulsbergen. - Braunschweig; Wiesbaden: Vieweg, 1992 (Aspects of mathematics: E; Vol. 18) NE: Aspects of mathematics / E
AMS subject classification: 11G, 11M, 14C, 14G, 14H, 14K, 19D, 19E, 19F. All rights reserved © Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig / Wiesbaden, 1992 Softcover reprint of the hardcover 1st edition 1992
Vieweg is a subsidiary company of the Bertelsmann Publishing Group International.
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 978-3-528-06433-4 DOl 10.1007/978-3-322-85466-7
ISBN 978-3-322-85466-7 (eBook)
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 11 14 17
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 Ellip
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