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Infinite descent on elliptic curves. (English) Zbl 0852.11028
In order to compute the Modell-Weil group $$E(K)$$ of an elliptic curve $$E$$ defined over an algebraic number field $$K$$ (often $$K= \mathbb{Q}$$), one first needs to calculate the rank of the curve. This is done by carrying out a 2-descent which, with some luck, gives a basis for the quotient group $$E(K)/ 2E (K)$$. From $$|E(K)/ 2E (K) |= 2^r$$, one immediately reads off the rank $$r$$. In most ‘reasonable’ cases – the discriminant of $$E$$ should not be too large – this part of the process is relatively straightforward. Once the rank is known, it usually is much harder to calculate a finite basis for $$E(K)$$. For this second part of the computation, the standard procedure is to use infinite descent, which is the process of obtaining a basis for $$E(K)$$, given a basis for $$E(K)/ mE (K)$$ for some $$m\geq 2$$. To this end one applies Zagier’s theorem in conjunction with explicit bounds for the difference of the logarithmic height $$h$$ and the canonical height $$\widehat {h}$$ of points $$P$$ on the curve [see: J. H. Silverman, Math. Comput. 55, 723-743 (1990; Zbl 0729.14026)].
In the present paper, the author describes a method by which the existing upper bound for $$h(P)- \widehat {h} (P)$$, established by Silverman in the paper cited above, can often be considerably improved. In many nontrivial examples, well-chosen from the existing literature, the author very convincingly illustrates the superiority of his approach.

##### MSC:
 11G05 Elliptic curves over global fields 11Y16 Number-theoretic algorithms; complexity 14H52 Elliptic curves 14Q05 Computational aspects of algebraic curves
PARI/GP; ecdata
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