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Extra info for Lectures on the Mordell-Weil Theorem

Example text

The group that 1,1 :::; 2C. r is generated by the elements, E r such Proof For each real number D > 0, we denote by r D the subgroup of r generated by the elements, E r with 1,1 :::; D. :l r mC = r, it suffices to check by induction on m that r mC C r(m-l)C, for m:2: 3. Let y E r mC; there is i and x E r with y = nx +'i. Then thus 1 Ixl:::; 2(lyl + hd) < (m+l)C. - 2 As m :2: 3, we have, (m + 1)/2C :::; (m - l)C, thus x E r(m-l)C. Proof of the Mordell- Weil theorem via the descent lemma. The lemma is applied to r = A(K), with Ixl 2 being the normalised height associated to some ample symmetric divisor class.

Suppose that one has constructed a basis El, ... , En of L n Vn . We shall complete this to a basis of L n Vn+1' The image of L n Vn+1 -+ Vn+1/Vn ~ Q is finitely 44 generated hence isomorphic to Z; let €n+l E LnVn + 1 be such that its image is a generator. Then €I, ••• , €n, ... is a basis of L. Complement. Let red) be the number of torsion points of A(K) of degree ~ d. Then: a) By Northcott's theorem, red) is finite and can be effectively bounded. b) If A is an elliptic curve without complex multiplication, we have red) ~ d3 / 2 , as d -+ 00.

Consider the map fy : A --t A X A', X I-t (x, y). We have fy = To,y 0 fo. From b), f;P = y E PicO(A). Hence y = f;P = foT;,yP, and since fo is a homomorphism, we have that is to say hy(x) from the lemma with a Whence the theorem. = hT>0,11 p(x,O) = hp(x, 0) - = (0, -y). But hp(x, 0) B(x, y), = 0 and B(x, y) = hp(x, y). Relation between quadratic and linear forms. Let c E Pic(A). We define c : A --t A' is a homomorphism.

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