regulator of an elliptic curveRegulatorOfAnEllipticCurve2006-11-08 21:02:202007-04-09 20:56:20Definitionelliptic curveelliptic regulatorheight matrix% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Cl}{\operatorname{Cl}}Let $E/\Rats$ be an elliptic curve, let $E(\Rats)$ be the group of rational points on the curve and let $\langle \cdot, \cdot \rangle$ be the N\'eron-Tate pairing:
$$\langle P,Q \rangle=\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q)$$
where $\hat{h}$ is the canonical height on the elliptic curve $E/\Rats$.
\begin{defn}
Let $E/\Rats$ be an elliptic curve and let $\{P_1,\ldots,P_r\}$ be a set of generators of the free part of $E(\Rats)$, i.e. the points $P_i$ generate $E(\Rats)$ modulo the torsion subgroup $E_{\operatorname{tors}}(\Rats)$. The {\bf height matrix} of $E/\Rats$ is the $r\times r$ matrix $H$ whose $ij$th component is $\langle P_i, P_j \rangle$, i.e.
$$H = (\langle P_i, P_j \rangle).$$
If $r=0$ then we define $H=1$.
\end{defn}
\begin{defn}
The {\bf \PMlinkescapetext{regulator}} of $E/\Rats$ (or the elliptic regulator), denoted by $\operatorname{Reg}(E/\Rats)$ or $R_{E/\Rats}$ is defined by
$$\operatorname{Reg}(E/\Rats)=\det(H)$$
where $H$ is the height matrix.
\end{defn}
Notice the similarities with the regulator of a number field. The regulator of an elliptic curve is the volume of a fundamental domain for $E(\Rats)$ modulo torsion, with respect to the quadratic form defined by the N\'eron-Tate pairing.