Abstract
Let C be a non-singular cubic curve given by an equation
with integer coefficients. We have seen that if C has a rational point (possibly at infinity), then the set of all rational points on C forms a finitely generated abelian group. So we can get every rational point on C by starting from some finite set and adding points using the geometrically defined group law.
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Notes
- 1.
But one has to be a little careful, since a silly equation such as x 3 = 1 has infinitely many solutions because y is arbitrary. Similarly, the equation \(x(x^{2} + xy - y) = 1\) has infinitely many solutions (1, y).
- 2.
Actually, we also need to check that W(X) is not the zero polynomial. We will verify this at the end of the proof.
- 3.
How do we know to choose exponents 9 and 18 and 65 in (5.1) and (5.2)? The answer is that initially we did not know. What we did was to write down the proof leaving the exponents as unknowns. Then, at the end, we could see which values would work. But there is nothing magical about 9, 18, and 65. Any larger values will also work, and if you redo the calculations with more care, you’ll find that there are smaller values that work, too.
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Silverman, J.H., Tate, J.T. (2015). Integer Points on Cubic Curves. In: Rational Points on Elliptic Curves. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-18588-0_5
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