Let $A$ be a principally polarised abelian variety defined over a number field $L$. The **canonical height** on $A$ is a function
$$
\hat{h}: A(L) \to \R_{ {}\ge0}
$$
defined on the Mordell-Weil group $A(L)$
which induces a positive definite quadratic form on $A(L)\otimes\R$.

One definition of $\hat{h}(P)$ is $\hat h(P)= h_{\mathrm{Kum}}(\sigma(P))$, where $\sigma \colon A \to K$ is the natural map from $A$ to its Kummer surface $K : = A/\{\pm 1\}$, and $h_\mathrm{kum} \colon K(L) \to \R$ the canonical height pairing associated to a theta divisor $\Theta$ on $K$, which can be defined by $$ h_{\mathrm{Kum}}(Q)=\lim_{n\to\infty} n^{-2}h_{\Theta}\bigl(nQ\bigr), $$ where $h_{\Theta}$ is the Weil height associated to a theta divisor $\Theta$ on $K$.

Related to the canonical height is the **height pairing**
$$
\langle-,-\rangle : A(L)\times A(L) \to \R
$$
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 $A(L)\otimes\R$, used in defining the regulator of $A/L$.

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**Knowl status:**

- Review status: beta
- Last edited by Raymond van Bommel on 2019-12-13 10:33:51

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