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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 $E/K$.

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