Nagell-Lutz theorem

This fact is related to: algebraic number theory
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This article or section of article is sourced from:Wikipedia

This theorem controls the abstract structure, as well as the concrete context of occurrence, of a group occurring in the following context: elliptic curve group

Statement

Let $f(x)=x^3+a{x}^2+bx+c$ with $a,b,c\in \mathbb{Z}$ such that the cubic curve $y^2=f(x)$ is nonsingular. Let $D$ denote the discriminant of $f$. Then, if $P$ is a point having finite order in the elliptic curve group, the following are true:

• If $p$ is not the point at infinity, then both its affine coordinates are integers
• The $y$-coordinate of $P$ is either zero, or a divisor of $D$

Relation with other results

Related results

• Mordell-Weil theorem
• Mazur's Theorem

External links

Statement links

• Wikipedia page
• Planetmath page