Simultaneous Representation of Integers by a Pair of Ternary Quadratic Forms—With an Application to Index Form Equations in Quartic Number Fields

Abstract

LetQ1, Q2∈Z[X, Y, Z] be two ternary quadratic forms andu1, u2∈Z. In this paper we consider the problem of solving the system of equations[formula]According to Mordell [12] the coprime solutions of[formula]can be presented by finitely many expressions of the formx=fx(p, q),y=fy(p, q),z=fz(p, q), wherefx, fy, fz∈Z[P, Q] are binary quadratic forms andp, qare coprime integers. Substituting these expressions into one of the equations of (1), we obtain a quartic homogeneous equation in two variables. If it is irreducible it is a quartic Thue equation, otherwise it can be solved even easier. The finitely many solutionsp, qof that equation then yield all solutionsx, y, zof (1). We also discuss two applications. In [8] we showed that the problem of solving index form equations in quartic numbers fieldsKcan be reduced to the resolution of a cubic equationF(u, v)=iand a corresponding system of quadratic equationsQ1(x, y, z)=u,Q2(x, y, z)=v, whereFis a binary cubic form andQ1, Q2are ternary quadratic forms. In this case the application of the new ideas for the resolution of (1) leads to quartic Thue equations which split over the same quartic fieldK. The second application is to the calculation of all integral points of an elliptic curve.