Diophantine Approximation and Abelian Varieties by Frits Beukers (auth.), Bas Edixhoven, Jan-Hendrik Evertse

By Frits Beukers (auth.), Bas Edixhoven, Jan-Hendrik Evertse (eds.)

The thirteen chapters of this ebook centre round the evidence of Theorem 1 of Faltings' paper "Diophantine approximation on abelian varieties", Ann. Math.133 (1991) and jointly supply an method of the facts that's obtainable to Ph.D-level scholars in quantity conception and algebraic geometry. each one bankruptcy relies on an educational lecture given by means of its writer ata detailed convention for graduate scholars, concerning Faltings' paper.

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Example text

To give a typical example, Tzanakis and de Weger [77] considered the Weierstrass equation y2 = x 3 - 4x + 1. They reduced it to some Thue equations, of which f( x, y) = x 4 -12x 2 y 2 - 8xy 3 +4y4 = 1 is a typical example. This leads to some equations where £1, £2, £3 is a fixed fundamental set of units of Q(t9) and iJ is a zero of f(x, 1). A suitable linear form estimate yields max(lall, la21, la31) < 1041 . Subsequently the basis reduction algorithm of Lenstra, Lenstra and Lovasz is applied. The first time yields an upper bound 72, the second time an upper bound 10.

Thus, ILp(Vq)\ ~ Bp for p = 1, N, q = 1, ... 14) , V N-t E IIk(B) n ZN. VI, We show that B satisfies (ii), (iii) and (i). Proof. • . ,liN. • , L N )1- 1 B 1 ••• BN. 4 this implies that Vt·· . liN ~ (B t ·· ·BN)-t. 14) implies that VN-l ~ 1. 15) liN ~ (B1 •• ·BN)-I. Note that AI··· AN = (AI··· A n )(n;;l), that ~1 ••• ~N = (At··· An )(n;l), and that = {k,k+2, ... ,n},O'N = {k+l,k+2, ,n}. 16) (AI ... AnAl . 15) this implies that rank(Ilk(B) n ZN) = N - 1. liN «:: Q{ A t 1 if Q{A) is sufficiently large.

Induction on n. For n = 1, the assertion is trivial. Let n > 1. There are an algebraic number field K and a finite set of places S of K, containing all archimedean places, such that G is contained in the group of S-units {x E K : IIxlIlI = 1 for v ~ S}. Define the linear form X o := atXt +.. ·+ anXn • Then {Xo, Xl, ... , X n } is in general position. Let x = (Xl' ... 8) and put Xo := 1. • xnll v = 1. Further, HK(X) = nvES II x lIlI. 8 implies that x E TI U·· ·UTh where Tt , . ,Th are proper hyperplanes of pn-I(K) independent of x.

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