The Decomposition of Primes in Torsion Point Fields
It is an historical goal of algebraic number theory to relate all algebraic extensionsofanumber?eldinauniquewaytostructuresthatareexclusively described in terms of the base ?eld. Suitable structures are the prime ideals of the ring of integers of the cons
- PDF / 3,061,656 Bytes
- 144 Pages / 436.611 x 666.4 pts Page_size
- 72 Downloads / 195 Views
Clemens Adelmann
The Decomposition of Primes in Torsion Point Fields
Springer
Author Clemens Adelmann Technical University Braunschweig Institute of Applied Mathematics Dept. of Applied Algebra Pockelsstr. 14 38 106 Braunschweig, Germany E-mail: [email protected]
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Adelmann, Clemens: The decomposition of primes in torsion point fields I Clemens Adelmann. - Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Singapore ; Tokyo : Springer, 2001 (Lecture notes in mathematics ; 1761) ISBN 3-540-42035-5
Mathematics Subject Classification (2000): 11-02, 1lR21, 1lR32, 1 lR09, 1 1G05, llF11, 13A50, 12Y05 ISSN 0075-8434 ISBN 3-540-42035-5 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, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms 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 Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH
O Springer-Verlag Berlin Heidelberg 2001 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 10836966 4113142-543210 - Printed on acid-free paper
Table of Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
Decomposition Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Foundations of Prime Ideal Decomposition . . . . . . . . . . . . . . . . . 5 2.2 Decomposition in Abelian Extensions . . . . . . . . . . . . . . . . . . . . . 11 2.3 Density Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3
Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Deļ¬ning Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Addition on Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Division Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Torsion Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 -adic Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Reduction Modulo p and L-Series . . . . . . . . . . . . . . . . . . . . . . . . .
25 25 26 28 30 32 35
4
Elliptic Modular Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Modular Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 M
Data Loading...
