By Prof. Jean-Pierre Serre (auth.), Martin Brown (eds.)
Professor Jean-Pierre Serre ist ein renommierter französischer Mathematiker am Collège de France, Paris.
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Extra info for Lectures on the Mordell-Weil Theorem
We associate to P E Pic(A x A') the normalised height A'(K). hp on A(K) x Proposition. Tbe beigbt hp is a quadratic form wbich is trivial on tbe two factors: hp(x,O) = 0, for x E A(K), hp(O, y) = 0 for y E A'(K). This follows from property a) of the Poincare divisor class. Let Bp be the bilinear form associated to hp. As (x, y) = (x, 0)+(0, V), we have hp(x, y) = hp(x, 0) + hp(O, y) + Bp«x, 0), (y, 0» = Bp«x,O),(y,O», therefore hp(x, y) is additive in x, y. We then have a canonical bilinear form on A(K) X A'(K).
Lemma 1. H A is a finitely generated subgroup of V , then, for any MER, there are only finitely many elements'x of A such that he('x) ::; M. Proof. We lift a set of generators of A to points xl, ... , Xn of A(K) with K finite over Q. 4). Lemma 2. Let F be a quadratic form in n variables xl, ... , Xn with real values. Suppose that for all MER there are only finitely many points (Xl, ... , xn) E zn with F(x) ::; M. Then F is positive and non-degenerate. Proof. : 0 on zn, hence on Qn, hence on Rn by continuity.
1). 3. Mordell-Weil theorem. Theorem. ) Let K be a number field, A an abelian variety over K, and r = A(K). Then r /nr is finite for all integers n 2: l. Proof. ) Let An be the kernel of the map 'Jr : x f-+ nx on A. 2), applied to the covering 'Jr, if P E A(K) then there is Q E A(K'), where [K' : K] ::; d(= n 2dim A) such that nQ = P, and K' /K is unramified outside a finite number of places. 1), the corresponding fields K' are finite in number; let L be a finite Galois extension of K containing all the K'.