Infinite descent on elliptic curves
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: \textit{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.
scientific article; zbMATH DE number 894664
scientific article