Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms

Abstract

1. Introduction. In [Z], Zagier describes several methods for explicitly computing (large) integral points on models of elliptic curves defined over Q. Here we are interested in the computation of all integral points on a given Weierstraß equation for an elliptic curve E/Q, but not merely by reducing the original diophantine equation to an equivalent finite set of Thue equations which are subsequently solved by elementary, algebraic or analytic methods (see [TdW] and [STz]). On the contrary, we adopt a more natural approach, one in which the linear (group) relation between an integral point and the generators of the free component of the Mordell-Weil group is directly transformed into a linear form in elliptic logarithms. This idea is not new; see [Ma, App. IV], [La, Ch. VI, §8], and [S1, Ch. IX, §5]. To make it work, that is to say, in order to produce upper bounds for the coefficients in the original linear (group) relation, one needs an effective lower bound for the linear form in elliptic logarithms. First to obtain such lower bounds were D. W. Masser [Ma, App. IV], in the case of elliptic curves with complex multiplication, and G. Wüstholz [Wu]; see also the bibliography in [H]. We felt that the recent result of N. Hirata-Kohno [H, Coroll. 2.16] should serve our purpose best. Unfortunately, this result, being rather more general than we required, though effective, is not completely explicit. At our request, S. David kindly undertook the highly non-trivial task of making explicit the special case we needed. It is S. David's result [D, Th. 2.1] that is applied here for the first time to provide explicit upper bounds for the coefficients in the linear (group) relation corresponding to a given Weierstraß equation. We shall show by example that these bounds may be reduced to manageable proportions.In the following sections we shall give a detailed description of the method referred to above. In the final section we present two examples, worked out in detail, of elliptic curves taken from the literature. Our choice

Prime Divisors of Sequences Associated to Elliptic Curves

“…Thus bounds from elliptic transcendence theory are applicable. We use Théorème 2.1 in [3] but see also [24] where an explicit version of David's Theorem appears on page 20. The nature of the bound is log |x n | v log n log log n,…”

Computing all integer solutions of a genus 1 equation

“…In case f (u, v) = 0 is a Weierstrass equation, a quartic equation of type v 2 = Q(u) for some quartic polynomial Q, or a general cubic elliptic equation, G(u, v) is constant with value 2; see [16], [21] and [20]. Now put…”

“…The group E(R), defined by y 2 = q(x), has the identity component E 0 (R), and in the real case-we remind the reader that in this case q(x) = 0 has three real roots e 1 > e 2 > e 3 -also the bounded component [16]). In the complex case-that is, when q(x) = 0 has a single real root-we have E 0 (R) = E(R), and φ is defined on the whole of E(R).…”

“…This method generalizes the so-called elliptic logarithm method-ellog for short-which, as a practical routine ('getting one's hands dirty', so to speak) for solving Weierstrass equations, was first described and applied by Stroeker and Tzanakis [16] and a little later, independently, by Gebel, Pethő and Zimmer [8]. These papers only differ in their general approach towards height estimates and software preference.…”

Lucas' square pyramid problem revisited

“…[37]) or on lower bounds for linear forms in elliptic logarithms (see e.g. [35]) to find all integer solutions (X, Y ) to Y 2 = X 3 − n 2 X and check to see which, if any, yield solutions to (1.3). To find positive integral solutions to (1.3), for all squarefree n up to some bound, say n ≤ N , it is computationally much more efficient however, to rely upon Theorem 2.2.…”