A simple practical higher descent for large height rational points on certain elliptic curv.pdf

a r X i v : m a t h / 9 9 0 4 1 7 2 v 1 [ m a t h .N T ] 3 0 A p r 1 9 9 9 A SIMPLE PRACTICAL HIGHER DESCENT FOR LARGE HEIGHT RATIONAL POINTS ON CERTAIN ELLIPTIC CURVES Allan J. MacLeod Dept. of Mathematics and Statistics University of Paisley High St. Paisley Scotland PA1 2BE (e-mail: macl-ms0@paisley.ac.uk) February 1, 2008 Abstract We consider the problem of determining a rational point (x, y), of infinite order, on the elliptic curve y2 = x3 + ax2 + bx. The well- known method of 4-descent has proven very effective, and we give explicit formulae for the various stages. For points of large height, however, there can still be a significant search to be done. We describe a very simple idea which essentially doubles the height range which can be dealt with in a reasonable time. Currently, the largest point found has height 51.15 or 102.3 (depending on which normalisation you use). The report is written in an algorithmic style and should be understandable by the interested amateur. 1 1 Introduction Many Diophantine problems can be reduced to determining points on ellip- tic curves. A fascinating example is from Bremner et al [3], where finding possible representations of n as n = (x+ y + z) ( 1 x + 1 y + 1 z ) is shown to be equivalent to finding points of infinite order on the elliptic curve E n : u2 = v3 + (n2 ? 6n? 3)v2 + 16nv The rank of E n can be estimated from the L-series of the curve, assuming the Birch Swinnerton-Dyer conjecture, but this just provides evidence of existence. For an actual representation, we need to explicitly find a rational point. If the curve has rank greater than 1, it is usually fairly simple to find a point by a trivial search. For rank 1 curves, however, this is often not feasible. The L-series computations can be extended to give an estimate for the height of a rational point of the curve. If this is large, then a simple search will take far too long. Recently, Silverman [7] produced a very nice and effective procedure to det