Computing integral points on elliptic curves

Abstract

Introduction. By a famous theorem of Siegel [S], the number of integral points on an elliptic curve E over an algebraic number field K is finite. A conjecture of Lang and Demyanenko (see [L3], p. 140) states that, for a quasiminimal model of E over K, this number is bounded by a constant depending only on the rank of E over K and on K (see also [HSi], [Zi4] [GSch]). The Siegel-Baker method (see [L3]) for the calculation of integer points on elliptic curves over K = Q requires some detailed information about certain quartic number fields. Computing these fields often represents a hard problem and, moreover, this approach does not seem to be appropriate. That is why in general all the results mentioned above cannot be used for the actual calculation of all integral points on an elliptic curve E over Q.However, there is another method suggested by Lang [L1], [L3] and further developed by Zagier [Za]. We shall work out the Lang-Zagier method and turn it into an algorithm for determining all integral points on elliptic curves E over Q using elliptic logarithms. The algorithm requires the knowledge of a basis of the Mordell-Weil group E(Q) and of an explicit lower bound for linear forms in elliptic logarithms. Compared to the Siegel-Baker

Solving the Diophantine equationy2=x(x2−n2)

“…We use the computational system KANT 2.5 and we obtain the following integral basis for the field K : [10,9,15,7]), ([2, 3 where θ = √ 3 + √ 5. We represent an algebraic integer of K , z = 3 (2,5,11,1), (11, 10, 1, 1), (11, 5, 2, 0) and (10, 11, 1, 0). The first two quadruples give the equation x 4 − 110 y 2 = 1.…”

“…Furthermore, if n is a perfect square, then, it is easily seen, that the rank of E n (Q) is zero. The integer points of E n can be determined by the elliptic logarithm method [11,17], if one knows a full set of generators for the group E n (Q), modulo torsion. Algorithms for finding such generators exist but are not guaranteed to give always an answer [5,10].…”

Brand Presence in Digital Space

“…These subroutines use the elliptic logarithm method (introduced in [14] and [24], see also [23,6]) for solving elliptic equations. These subroutines use the elliptic logarithm method (introduced in [14] and [24], see also [23,6]) for solving elliptic equations.…”

Solving elliptic diophantine equations: the general cubic case

“…There are very many examples available in the literature. See for instance [ST94], [GPZ94], [Sm94], [St95], [BST97], [SdW97], [ST97], and [HP98].…”