Elliptic Curves Notes from Postgraduate Lectures Given in Lausanne 1
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326 Alain Robert
Elliptic Curves Notes from Postgraduate Lectures Given in Lausanne 1971/72
Springer-Verlag Berlin Heidelberg New York Tokyo
Author Alain Robert Universite de Neuchitel, Institut de Mathematiques Chantemerle 20, 2000 Neuchitel, Switzerland
1st Edition 1973 2nd Corrected Printing 1986
Mathematics Subject Classification (1980): 12835, 12B37, 14G 10, 14H 15, 14H45, 32G 15 ISBN 3-540-06309-9 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-06309-9 Springer-Verlag New York Heidelberg Berlin Tokyo 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 -Verwertungsgesellschaft Wort", Munich. C by Springer-Verlag Serren Heidelberg 1973 Printed in Germany Printing and binding: Seltz Offsetdruck, Hemsbach/8ergstr.
2146/3140-543210
NOTATIONS AND
CON V E NT ION S
We have used the usual letters for the basic sets of numbers N (natural integers 0,1,2, ... ), ~
~
(ring of rational integers),
(field of rational numbers), R (field of real numbers), [ (field
of complex numbers), F q (finite field with q elements). As a rule, we denote by AX the multiplicative group of units (invertible elements) in a ring A. In formulas, the cypher 1 always represents the number one (except in log x ... so that in one occurence I have used log -1 to avoid ambiguities). Also ~(x) =e
2rrix
(normalized exponential).
In a theorem, I list properties under Latin letters a), b), ... keeping
i), ii), ... for equivalent properties, but the meaning is
always clear by the context. The following system has been adopted for cross-references. All theorems, propositions, corollaries, lemmas, remarks, definitions, formulas, errata, ... are numbered in one sequence. Such a cypher as (nL3.4) refers to the item (3.4) of chapter III, i.e. the fourth numbered in section 3. (This happens to be a lemma 3.J" From inside chapter III we would refer to (3.4) (in section 3, sometimes simply to lemma 3
this last system of numeration has not been used systema-
tically, but only when it can be more suggestive locally).
ERRATA Pages nwnbers refer to bottom pages numbers. p.73
p.74 p.102
p.183 p.23l p.234
line 2 from bottom. It is not true that ~(n) is divisible by 691 for nearly all integers. For example, when p is prime, -c(p) :: 1 + pil -mod 691 and the set of primes p such that t(p) is divisible by 691 has density 1/690 according to Dirichlet's theorem. But ~(n) = Oil(n) mod 691 and Oil(n) is divisible by 691 for "nearly all n" in the sense of the naive density on the set of integers ••• cf. J.-P. Serre, C.R.Acad.Sc.Paris 279 (1974), pp.679-682. Since Weil's conjectures have been proved by P. Deligne, Ramanujan's conjecture is true. In the case of a double point, the isomorphism of the sMOoth
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