Computing Canonical Heights on Elliptic Curves

The theory of heights is an extremely important tool in the study of the arithmetic of elliptic curves. For $E/K$ an elliptic curve over a number field, the canonical height is a quadratic form $\hat{h}: E(K) \rightarrow \mathbb{R}$ with several desirable properties and far-reaching theoretical applications, playing a role in the celebrated Mordell-Weil Theorem, and constituting a term (the regulator) in the Birch-Swinnerton-Dyer conjecture, one of the most important open problems in mathematics. The utility of heights extends to the computational theory of elliptic curves, where we aim to algorithmically compute the arithmetic invariants of a given elliptic curve $E$. This can be used to aid theory by providing evidence for conjectures (for example, searching for elliptic curves with large rank), with databases such as LMFDB containing millions of elliptic curves over $\mathbb{Q}$ and small number fields.