Solving Elliptic Diophantine Equations by Estimating Linear Forms in Elliptic Logarithms
نویسنده
چکیده
In order to compute all integer points on a Weierstraß equation for an elliptic curve E/Q, one may translate the linear relation between rational points on E into a linear form of elliptic logarithms. An upper bound for this linear form can be obtained by employing the Néron-Tate height function and a lower bound is provided by a recent theorem of S. David. Combining these two bounds allows for the estimation of the integral coefficients in the group relation, once the group structure of E(Q) is fully known. Reducing the large bound for the coefficients so obtained to a manageable size is achieved by applying a reduction process due to de Weger. In the final section two examples of elliptic curves of rank 2 and 3 are worked out in detail.
منابع مشابه
Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms
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 subse...
متن کاملSolving Elliptic Diophantine Equations Avoiding Thue Equations and Elliptic Logarithms
This research was supported by the Netherlands Mathematical Research Foundation SWON with nancial aid from the Netherlands Organization for Scienti c Research NWO. We determine the solutions in integers of the equation y = (x + p)(x + p) for p = 167, 223, 337, 1201. The method used was suggested to us by Yu. Bilu, and is shown to be in some cases more efficient than other general purpose method...
متن کاملOn a Few Diophantine Equations, in Particular, Fermat’s Last Theorem
This is a survey on Diophantine equations, with the purpose being to give the flavour of some known results on the subject and to describe a few open problems. We will come across Fermat’s last theorem and its proof by Andrew Wiles using the modularity of elliptic curves, and we will exhibit other Diophantine equations which were solved à la Wiles. We will exhibit many families of Thue equation...
متن کاملAsymptotically Fast Discrete Logarithms in Quadratic Number Fields
This article presents algorithms for computing discrete logarithms in class groups of quadratic number elds. In the case of imaginary quadratic elds, the algorithm is based on methods applied by Hafner and McCurley HM89] to determine the structure of the class group of imaginary quadratic elds. In the case of real quadratic elds, the algorithm of Buchmann Buc89] for computation of class group a...
متن کاملElliptic binomial diophantine equations
The complete sets of solutions of the equation (n k ) = (m ` ) are determined for the cases (k, `) = (2, 3), (2, 4), (2, 6), (2, 8), (3, 4), (3, 6), (4, 6), (4, 8). In each of these cases the equation is reduced to an elliptic equation, which is solved by using linear forms in elliptic logarithms. In all but one case this is more or less routine, but in the remaining case ((k, `) = (3, 6)) we h...
متن کامل