A Diophantine problem on elliptic curves

Tubbs, Robert

doi:10.1090/S0002-9947-1988-0957074-8

A Diophantine problem on elliptic curves
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by Robert Tubbs

Trans. Amer. Math. Soc. 309 (1988), 325-338

DOI: https://doi.org/10.1090/S0002-9947-1988-0957074-8

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Abstract:

This paper examines simultaneous diophantine approximations to coordinates of certain points on a product of elliptic curves. Specifically, let $\wp (z)$ be a Weierstrass elliptic function with algebraic invariants and complex multiplication. Suppose that $\beta$ is cubic over the "field of multiplications" of $\wp (z)$ and that $u \in \mathbb {C}$ such that $\zeta = (\wp (u), \wp (\beta u), \wp ({\beta ^2}u))$ is defined. We study approximations to $\zeta$ by points which lie on curves defined over $\mathbb {Z}$.

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