Regulator of an abelian variety (awaiting review)

The regulator of an abelian variety $A$ defined over a number field $K,$ denoted $\operatorname{Reg}(A/K)$, is the volume of $A(K)/A(K)_{tor}$ with respect to the height pairing $\langle -,-\rangle$ associated to the canonical height $\hat{h}$, i.e. $\langle P,Q\rangle = \frac{1}{2}(\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q))$.

If the Mordell-Weil group $A(K)$ has rank $r$ and $P_1, \ldots, P_r \in E(K)$ generate $A(K)/A(K)_{tor}$, then \[ \operatorname{Reg}(A/K) = \left|\det (\langle P_i, P_j \rangle )_{1\leq i,j \leq r}\right|, \] which is independent of the choice of generators.

Special cases are when $A(K)$ has rank $0$, in which case $A(K)/A(K)_{tor}=0$ and $\operatorname{Reg}(A/K)=1$, and when $A(K)$ has rank $1$, in which case $\operatorname{Reg}(A/K)$ is equal to the canonical height $\hat{h}(P)$ of a generator $P$.

This is a generalisation of the regulator of an elliptic curve.