Abstract
An element P of any group is said to have order m if
but \(m'P\neq \mathcal{O}\) for all integers 1 ≤ m′ < m. If such an m exists, then P has finite order, otherwise it has infinite order. We begin our study of points of finite order on cubic curves by looking at points of order two and order three. As usual, we will assume that our non-singular cubic curve is given by a Weierstrass equation
and that the point at infinity \(\mathcal{O}\) is taken to be the zero element for the group law.
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Notes
- 1.
However, there are two caveats. First, as with the case of rational, real, or complex numbers, we must assume that the cubic polynomial \(x^{3} + ax^{2} + bx + c\) does not have a double root in the algebraic closure of the finite field. Second, the formulas do not work for fields of characteristic 2. The problem occurs when we try to go from a general cubic equation to an equation of the form y 2 = f(x). This transformation requires dividing by 2 and completing the square; see Section 1.3 To work with cubic equations in characteristic 2, one uses more general Weierstrass equations of the form \(y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}\).
- 2.
- 3.
We remark that the resultant of f(X) and ϕ(X) is actually D 2, so general theory only predicts an equation of the form \(Ff+\varPhi \phi = D^{2}\).
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Silverman, J.H., Tate, J.T. (2015). Points of Finite Order. In: Rational Points on Elliptic Curves. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-18588-0_2
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