Frobenius Distributions in GL2-Extensions Distribution of Frobenius
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504 Serge Lang Hale Trotter
Frobenius Distributions in GL2 -Extensions Distribution of Frobenius Automorphisms in of the Rational Numbers
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Springer-Verlag Berlin· Heidelberg · NewYork 1976
Authors Serge Lang Mathematics Department Yale University New Haven, Connecticut 06520 USA Hale Freeman Trotter Fine Hall Princeton University Princeton, New Jersey 08540 USA
Library of Congress Cataloging in Publication Data
Lang, Serge, 1927-
Frobenius distributions in GL2-extensions.
(Lecture notes in mathematics ; 504) Bibliography: p. Includes index. 1. Probabilistic number theory. 2. Galois theory. 3. Field,extensions (Mathematics) 4. Numbers, Rational. I. Trotter, Hale F., joint author. II. Title. III. Series: Lecture notes in mathematics (Berlin) ; 504. QA3.L28 no. 504 [QA241.7] 510'.8s [512'.7] 75-45242
AMS Subject Classifications (1970): lOK99, 12A55, 12A 75, 33A25 ISBN 3-540-07550-X Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-07550-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 1976 Printed in Germany Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
ACKNOWLEDGMENTS Both Lang and Trotter are supported by NSF grants. The final draft of this monograph was written while Lang was at the Institute for Advanced Study, whose hospitality we appreciated. We also thank Mrs. Helen Morris for a superb job of varityping the final copy.
INTRODUCTION We are interested in a distribution problem for primes related to elliptic curves, but which can also be described solely in terms of the distribution of Frobenius elements in certain Galois extensions of the rationals. We therefore first describe this situation, and then indicate its connection with elliptic curves. Let K be a Galois (infinite) extension of the rationals, with Galois group G. We suppose given a representation
which we assume gives an embedding of G onto an open subgroup of the product, taken over all primes
e.
We let
be the projection of P on the e-th factor. We assume that there is an integer !!l such that if p is a prime and p
r !!le,
then p is unramified in
rr
or in other
words, the inertia group at a prime of K above p is contained in the kernel of
Pf Then the Frobenius class a p is well defined in the factor group
Ge and
Pe (ap)
=
G/Ker
Pe '
has a characteristic polynomial which we assume of the form
e,
We assume that t p is an integer independent of and call t p the trace of Frobenius, Finally we assume that the roots of the characteristic polynomial have absolute value
vp,
and are complex conjugates
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