By Noboru Aoki, Professor Tetsuji Shioda (auth.), Michael Artin, John Tate (eds.)

ISBN-10: 0817631321

ISBN-13: 9780817631321

**Read or Download Arithmetic and Geometry: Papers Dedicated to I.R. Shafarevich on the Occasion of His Sixtieth Birthday Volume I Arithmetic PDF**

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**Additional info for Arithmetic and Geometry: Papers Dedicated to I.R. Shafarevich on the Occasion of His Sixtieth Birthday Volume I Arithmetic**

**Example text**

13) Suppose, first, that. 11) i' 0, l2 = ma M such that = 0. 15) since both sides have degree ::; 4. Hence G(T) is a perfect square, a case we have already eliminated. ) (say) for which l2 = 0, since this is a single condition. Then m 3 '/:- 0 and, by homogeneity, we may take m3 = 1. Similarly, there is a pair (L, M) = (L2, M2) for which m3 = 0 and l2 = 1. 2_17) =degree 1 = T 3 +degree 2 = T 2 + degree 1 = degree 2. 18) for some w = w(u, v) E k. Put 'U = 1, v = ±1, where we consider both choices of sign together for the moment.

Elation fore is of degree 5. Hence T2 = 0 and at = a2 E k. There are now two cases: (i) d 1 + d2 = 0. 1} E 29. (ii) d 1 = d2. 9) ~={a, a}= 2{o,a} E 29. 2. 7) does not hold. 12) L(9)p = M(e). 7) does not hold. Hence the set of pairs {L, M} is a 2-dimensional k-linear space. 13) Suppose, first, that. 11) i' 0, l2 = ma M such that = 0. 15) since both sides have degree ::; 4. Hence G(T) is a perfect square, a case we have already eliminated. ) (say) for which l2 = 0, since this is a single condition.

In 1928 Weil [5] in his thesis generalized Mordell's result to abelian varieties of any dimension and to any algebraic number field as ground field. At the same time, Weil [6] gave a very simple and elegant proof of Mordell's original result. I observed some time ago that Wcil's simple proof admits a further simplification. He uses the explicit form of the duplication and addition theorems on the abelian variety, but these can be avoided by, roughly speaking, the observation that every element of a cubic field-extension k(O) of a field k can be put in the shape (aO + b)/(cO +d)( a, b, c, dE k).