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Survey Article—Diophantine Equations with Special Reference to Elliptic Curves
Author(s) -
Cassels J. W. S.
Publication year - 1967
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s1-42.1.183-s
Subject(s) - diophantine equation , mathematics , prime (order theory) , elliptic curve , line (geometry) , principal (computer security) , prime number , confusion , arithmetic progression , algebra over a field , arithmetic , combinatorics , pure mathematics , calculus (dental) , discrete mathematics , geometry , computer science , medicine , psychology , dentistry , psychoanalysis , operating system
(I). The middle line of equation (6.3) on p. 210 should read : “ =0 or 1 if deg D = 0 ”. The D with l (D) = 1 and deg D = 0 are, of course, precisely the principal divisors. (II). There is an unfortunate confusion in the first six lines of p. 278. To prove Šafarevič's theorem one must consider not III m but ц m , where ц is given by (27.1). Since this group is also finite, the argument works with this alteration. The finiteness of ц m for every m follows at once from the finiteness of ц q for every prime q . And this is a consequence of Theorem 7.1 of Cassels (1962a) [ cf . also the reformulation on p. 153 of Cassels (1964b)] combined with the finiteness of the Selmer group S ( q ) . (III). Minor misprints. In the second footnote on p. 254 replace the first “ = ” by “ ≠ ”. On p. 284, line 7 replace “ g ' ” by “ g ” and the second “ ½ ” by “ 0 ”. (IV). The references Serre (1964b) and Tate (1964a) are now generally available in English in: Arithmetic algebraic geometry , Proceedings of a conference held at Purdue University, December 5–7, 1963 (Harper and Row, New York, 1966).