Icosahedral Galois Representations
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654 Joe P. Buhler
Icosahedral Galois Representations
Springer-Verlag Berlin Heidelberg New York 1978
Author Joe P. Buhler Mathematics Department The Pennsylvania State University University Park, PA 16802/USA
Library of Congress Cataloging in Publication Data
Buhler, Joe P 1950Icosahedral galois representations. (Lecture notes in mathematics; 654) Bibliography: p. Includes index. 1.
Algebraic number theory.
2
e
-
Saj.o-i.s theory.
3. Automorphic forms. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 654. QA3.128 no. 654 [Q,A247] 510' .Bs [512'.74] 78-9714 1SBN 0-387-08844-x
AMS Subject Classifications (1970): 12A55, 12BlO, lODlO
ISBN 3-540-08844-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08844-X Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
TABLE OF CONTENTS
Introduction Chapter 1: Lifting projective representations
6
Chapter 2: Local primitive galois representations
18
Chapter 3: Two-dimensional representations over Q
36
Chapter 4: The L-series of an icosahedral representation
49
Chapter S: Classical modular forms of weight one
70
Chapter 6: An icosahedral form
80
Bibliography
95
Appendix 1: The sextic resolvent
98
Appendix 2: Extensions of Q of degree 5 .........•........••.••...••. 100 S Appendix 3: The A4 extension of Q ...••.......................•.•...•• 103 2 Appendix 4: S4 extensions of Q unramified outside 2 and 5 108 Appendix S: Algorithms
115
Appendix 6: Fourier expansions at arbitrary cusps
125
Appendix 7: The holomorphy of A L-series .•••.......•....•••••.•..•.. 132 5 Index ......••.•.....•.•........................•..........•....•.•••.. 142 TABLES 3.1: Ramified primes in AS extensions .•..•.......•......•••••.•.•.... 46 3.2: Some low icosahedral conductors ..•.••...•....••.....•••••••.... 47 3.3: Primitive extensions of Q2 . 48 . 64 4.1 : Frobenii in an AS field 4.2: Decomposition of primes in an A field •...•.••.....••••••••...• 65 5 66 4.3: Some norms ••••••..•..........•••.••.....•..•••....•.•.••••••..•• 4.4: Values of lb ••••••••••••••••••••••••••••••••••••••••••••••••••• 68 4.5: An icosahedral L-series ....•...................•...•.•..•.••... 69 5.1: Dihedral cusp forms .•..••......••...........•.......•.•••...•.. 78 S.2: Forms of weight 2 ..........•••..••...........•.......••••.•.... 79 App4.1: Some dihedral extensions of Q •.•....•••........•...•.•..••• 114 2 135 App7.1: Table of SL?CF ...•.•.........••.•......•••••••.. S) A
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