Heights and the Mordell–Weil theorem

For a rational affine point with x=a/b in lowest terms and b positive, define the logarithmic x-height as log max(|a|,b), and set the height of O to zero. A number field uses the absolute logarithmic projective height, with local weights normalized by its degree over Q. Bounded height gives finitely many points in a fixed number field by Northcott's theorem. The fixed field or bounded-degree hypothesis is essential: over an algebraic closure there are infinitely many torsion points of canonical height zero.

We use the Silverman normalization: the canonical height is one half the limit of four to the minus n times the x-height of 2ⁿP. Thus it differs from half the naive height by a bounded function. Some sources and software instead call the unhalved limit the height. Both are valid conventions, but a numerical comparison or a regulator formula must account for the factor two. An additive change of bounded size disappears in the limit; a multiplicative normalization does not.

Mordell–Weil states that E(K) is a finitely generated abelian group when K is a number field. Hence it is a free abelian group of finite rank plus a finite torsion subgroup. Weak Mordell–Weil first proves that E(K)/nE(K) is finite for an integer n greater than one. That statement alone is insufficient: the additive group Q has trivial quotient by nQ but is not finitely generated. A separate arithmetic compactness argument must control repeated division.

Choose finitely many coset representatives. Any point can be written as nQ plus one of them. The quadratic height and its pairing show that Q has smaller height once the starting point lies beyond a bound depending on those representatives. Repeating this reduction reaches a bounded-height set, which is finite by Northcott. The representatives together with that finite set generate E(K). This is an existence argument for a finite basis, not automatically a terminating rank algorithm for every input curve.

The canonical height is nonnegative and vanishes exactly on torsion points over a number field. The pairing is positive definite on the free part, so a nonsingular positive Gram matrix can certify independence of a proposed collection of points. Independence gives a lower bound for rank; it does not establish that those points span the entire free group. Even a full-rank subgroup can have finite index, affecting a regulator by the square of that index.

Finite approximations to the height need error bounds. If a global duplication inequality bounds the absolute difference between h_x(2Q) and four h_x(Q) by C, telescoping gives an error at most C divided by six times four to the n for the half-limit approximant. The constant must be proved for the chosen curve and model. A few stable decimal digits without such control are useful evidence but do not certify a zero height or rule out an omitted independent point.