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The regulator of an elliptic curve $E$ defined over a number field $K$, denoted $\operatorname{Reg}(E/K)$, is the volume of $E(K)/E(K)_{tor}$ 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 $E(K)$ has rank $r$ and $P_1, \ldots, P_r \in E(K)$ generate $E(K)/E(K)_{\mathrm{tor}}$, then \[ \operatorname{Reg}(E/K) = \left|\det (\langle P_i, P_j \rangle )_{1\leq i,j \leq r}\right|, \] which is independent of the choice of generators.

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

The canonical height used to define the regulator is usually normalised so that it is invariant under base change. Note that the regulator which appears in the Birch Swinnerton-Dyer conjecture is with respect to the non-normalised height.

Knowl status:
  • Review status: reviewed
  • Last edited by Andrew Sutherland on 2020-10-13 16:46:35
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