Survey Article-Diophantine Equations with Special Reference to Elliptic Curves

Abstract

(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 IIIm but цm, where ц is given by (27.1). Since this group is also finite, the argument works with this alteration. The finiteness of цm for every m follows at once from the finiteness of цq 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).