A Diophantine Problem on Elliptic Curves

This paper examines simultaneous diophantine approximations to coordinates of certain points on a product of elliptic curves. Specifically, let p(z) be a Weierstrass elliptic function with algebraic invariants and complex multiplication. Suppose that 0 is cubic over the "field of multiplications" of p(z) and that u 6 C such that f = (p{u), p(/3u), p(/32u)) is defined. We study approximations to c by points which lie on curves defined over Z. In this paper we investigate how closely, in an appropriate sense, curves defined over Z can come to certain points in C3 whose coordinates are given as values of elliptic functions. We show that integral polynomials which define the curve locally cannot both have moduli at the point, which are small in terms of the degree and height of the polynomials. This study was motivated by a desire to provide an elliptic analogue to W. D. Brownawell's generalization [3] of A. O. Gelfond and N. I. Feldman's measure for the algebraic independence of a13 and a& for a, 0 algebraic with a / 0,1 and 0 cubic over Q, [7]. Let p(z) be a Weierstrass elliptic function satisfying the Weierstrass equation p'(z)2 = 4p3(z) g2p(z) g3 and with lattice of periods Sf = wyZ + u2l. We assume throughout this paper that g2 and g$ (the invariants of p(z)) are algebraic. Additionally, we assume that p(z) has complex multiplication, in which case r = <jj2/u)y is a quadratic irrationality and KT = Q(r) is called the field of multiplications for p(z). For a polynomial P over C, in one or several variables, let dP denote the total degree of P, dxP the partial degree of P with respect to x, and d*P = max{l, dxP}. The height of P, ht P, is defined to be the maximum absolute value of the coefficients of P, and t(P) = dP + log MP is called the size of P. Moreover, for a pair of polynomials Py (x,y) and P2(x,y,z) we define several quantities which appear below. Namely, let A = d*xPy(dyP2 + d*P2) + d*yPy(dxP2 + d'zP2), B = d*zP2(d;Py + foghtPi + l0g(l + d^Pi)) + dyPy (dyP2 + d*zP2 + loght P2 + log(l + dP2)) + dyP2 loght Pi, and define a real number r by logr = A2^^d*.Pid;P2. The main result of this paper is the following theorem. Received by the editors June 6, 1986 and, in revised form, June 23, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 11J89; Secondary 11G99. Research supported in part by a grant from the National Science Foundation. ©1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page 325 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use