Let $E$ be an elliptic curve defined over a number field $K$. The **canonical height** on $E$ is a function
$$
\hat{h}: E(K) \to \R_{ {}\ge0}
$$
defined on the Mordell-Weil group $E(K)$
which induces a positive definite quadratic form on $E(K)\otimes\R$.

One definition of $\hat{h}(P)$ is $$ \hat h(P)=\lim_{n\to\infty} n^{-2}h\bigl(x(nP)\bigr), $$ where $h(x)$ is the Weil height of $x\in K$. This definition gives the non-normalised height. A normalised height which is invariant under base-change is given by $$ \frac{1}{[K:\Q]} \hat{h}(P). $$

Related to the canonical height is the **height pairing**
$$
\langle-,-\rangle : E(K)\times E(K) \to \R
$$
defined by $\langle P,Q\rangle = \frac{1}{2}(\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q))$, which is a positive definite quadratic form on $E(K)\otimes\R$, used in defining the regulator of $E/K$.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by John Cremona on 2020-10-07 11:49:37

**History:**(expand/hide all)

- 2020-10-07 11:49:37 by John Cremona (Reviewed)
- 2019-02-08 12:38:09 by John Cremona (Reviewed)

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