Cubic Fields and Mordell Curves

Abstract

The purpose of this article is to describe a relationship between several cubic polynomials and elliptic curves, and show a clearer view on it than that in the former half of our previous work [Mi-2003]. For a monic irreducible cubic polynomial P(u) in u over ℚ, the curve E = E(P(u)) defined by the equation w³ = P(u) is an elliptic curve whose j-invariant is equal to 0. We describe the set E[ℚ] of all rational points of E over ℚ by use of a root ξ of P(u) as

$$ \mathcal{W}\left( \xi \right) = \left\{ {\alpha = q\xi + r\left| {N_{K/\mathbb{Q}} } \right.\left( \alpha \right) = 1,q,r \in \mathbb{Q}} \right\}. $$

Then we show that the short form of E is a Mordell curve, y² = x³ + k, with a certain rational number k determined by the coefficients of P(u). It is also pointed out that E(P(u)) is essentially dependent on the polynomial P(u) rather than the cubic field ℚ(ξ) even though E[ℚ] is completely described by the subset W(ξ) of the cubic field.

The author was partly supported by the Grant-in-Aid for Scientific Research (C) (2) No. 14540037, Japan Society for the Promotion of Science, while he prepared this work.