# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] B.J. Birch and H.P.F. Swinnerton-Dyer, Notes on elliptic curves I, J. Reine Angew. Math. 212 (1963), 7-25. · Zbl 0118.27601 [2] J.P. Buhler, B.H. Gross and D.B. Zagier, On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3, Math. Comp. 44 (1985), 473-481. JSTOR: · Zbl 0606.14021 [3] A. Bremner, On the equation $$Y^2=X(X^2+p)$$ , in Number theory and applications (R.A. Mollin, ed.), Kluwer, Dordrecht, 3-23, 1989. [4] A. Bremner and J.W.S. Cassels, On the equation $$Y^2=X(X^2+p)$$ , Math. Comp. 42 (1984), 257-264. JSTOR: · Zbl 0531.10014 [5] J.W.S. Cassels, Lectures on elliptic curves , LMS Student Texts, Cambridge University Press, 1991. · Zbl 0752.14033 [6] ——–, Rational quadratic forms , LMS Monographs, Academic Press, London, 1978. [7] ——–, Introduction to the geometry of numbers , Springer-Verlag, 1959. · Zbl 0086.26203 [8] H. Cohen, A course in computational algebraic number theory , GTM 138 -1993. · Zbl 0786.11071 [9] J.E. Cremona, Algorithms for modular elliptic curves , Cambridge University Press, 1992. · Zbl 0758.14042 [10] V.A. Dem’janenko, An estimate of the remainder term in Tate’s formula , Mat Zametki 3 (1968), 271-278, in Russian. · Zbl 0161.40601 [11] J. Gebel and H.G. Zimmer, Computing the Mordell-Weil group of an elliptic curve over $$Q$$ , in Elliptic curves and related topics (H. Kisilevsky and M. Ram. Murty, ed.), CRM Proceedings and Lecture Notes Volume 4, Amer. Math. Soc., 1994. · Zbl 0809.14024 [12] J. Gebel, A. Pethő and H.G. Zimmer, Computing integral points on elliptic curves , Acta Arith., · Zbl 0816.11019 [13] D. Husemoller, Elliptic curves , Springer-Verlag, 1987. · Zbl 0605.14032 [14] T.J. Kretschmer, Construction of elliptic curves with large rank , Math. Comp. 46 (1986), 627-635. JSTOR: · Zbl 0593.14022 [15] J.-F. Metre, Construction d’une courbe elliptique de rang $$\geq12$$ , C.R. Acad. Sci. Paris 295 (1982), 643-644. · Zbl 0541.14027 [16] C. Batut, D. Bernardi, H. Cohen and M. Olivier, User’s guide to PARI-GP (version 1.38.62), 1994. [17] H.A. Priestley, Introduction to complex analysis , Oxford University Press, 1985. · Zbl 0569.30001 [18] C.L. Siegel, Lectures on the geometry of numbers , Springer-Verlag, 1988. · Zbl 0691.10021 [19] S. Siksek, Descents on curves of genus 1, Ph.D. thesis, Exeter University, 1995. · Zbl 0852.11028 [20] J.H. Silverman, The difference between the Weil height and the canonical height on elliptic curves , Math. Comp. 55 (1990), 723-743. JSTOR: · Zbl 0729.14026 [21] ——–, The arithmetic of elliptic curves , GTM 106 -1986. · Zbl 1205.65063 [22] ——–, Computing heights on elliptic curves , Math. Comp. 51 (1988), 339-358. JSTOR: · Zbl 1042.11036 [23] N.P. Smart, $$S$$-integral points on elliptic curves , Proc. Camb. Phil. Soc. 116 (1994), 391-399. · Zbl 0817.11031 [24] N.P. Smart and N.M. Stephens, Integral points on elliptic curves over number fields , · Zbl 0881.11054 [25] R.J. Stroeker and J. Top, On the equation $$Y^2=(X+p)(X^2+p^2)$$ , Rocky Mountain J. Math. 24 (1994), 1135-1161. · Zbl 0810.11038 [26] R.J. Stroeker and N. Tzanakis, Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms , Acta Arith. 67 (1994), 177-196. · Zbl 0805.11026 [27] H.G. Zimmer, On the difference of the Weil height and the Neron-Tate height , Math. Z. 147 (1976), 35-51. · Zbl 0303.14003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.