Heights & Mordell–Weil Lattices¶

The canonical height \(\hat h\) is the natural quadratic height on the Mordell–Weil group.

For non-torsion \(P\),

\[ \hat h(nP)=n^2\hat h(P). \]

The associated bilinear pairing is

\[ \langle P,Q\rangle= \frac{1}{2} \left( \hat h(P+Q)-\hat h(P)-\hat h(Q) \right). \]

For independent points modulo torsion, this pairing gives a positive-definite Gram matrix.

Lattice model¶

A rank-\(r\) subgroup with basis \(P_1,\dots,P_r\) becomes a Euclidean lattice through the height pairing.

The Gram matrix is

\[ G_{ij}=\langle P_i,P_j\rangle. \]

Its determinant is related to the regulator of the chosen subgroup.

Two bases can span the same subgroup while having very different numerical quality.

A poor basis may contain huge combinations and make:

more difficult.

Lattice reduction seeks a better basis of the same subgroup.

Canonical-height computations are high-precision real calculations.

For large-height points or nearly dependent bases, low precision can produce unstable determinants/eigenvalues.

Increase precision before drawing geometric conclusions.

A numerical Cholesky/Gram test is a screen.

It can suggest that points are independent, but Rank Hunter's rigorous lower bound uses the exact certificate path.

This separation matters most in high rank, where condition numbers can be extreme.

Lattices are useful for choosing combinations of known points as search anchors.

Examples include:

These choices can make transformed point coordinates much smaller.

Whether they actually expose a new rational point remains a search question.

If a basis is unsaturated, the lattice describes a sublattice of the true Mordell–Weil lattice.

Saturation can replace it with a larger same-rank subgroup and change regulator/index relationships.

A nonzero numerical regulator is not, by itself, the Rank Hunter proof that rank equals the matrix dimension.

The exact points and independence certificate provide the lower bound; the lattice describes their geometry.