# regulator of an elliptic curve

Let $E/\mathbb{Q}$ be an elliptic curve, let $E(\mathbb{Q})$ be the group of rational points on the curve and let $\langle\cdot,\cdot\rangle$ be the Néron-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 $E/\mathbb{Q}$.

###### Definition 1.

Let $E/\mathbb{Q}$ be an elliptic curve and let $\{P_{1},\ldots,P_{r}\}$ be a set of generators of the free part of $E(\mathbb{Q})$, i.e. the points $P_{i}$ generate $E(\mathbb{Q})$ modulo the torsion subgroup $E_{\operatorname{tors}}(\mathbb{Q})$. The height matrix of $E/\mathbb{Q}$ 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 $H=1$.

###### Definition 2.

The of $E/\mathbb{Q}$ (or the elliptic regulator), denoted by $\operatorname{Reg}(E/\mathbb{Q})$ or $R_{E/\mathbb{Q}}$ is defined by

 $\operatorname{Reg}(E/\mathbb{Q})=\det(H)$

where $H$ is the height matrix.

Notice the similarities with the regulator of a number field. The regulator of an elliptic curve is the volume of a fundamental domain for $E(\mathbb{Q})$ modulo torsion, with respect to the quadratic form defined by the Néron-Tate pairing.

 Title regulator of an elliptic curve Canonical name RegulatorOfAnEllipticCurve Date of creation 2013-03-22 16:23:24 Last modified on 2013-03-22 16:23:24 Owner alozano (2414) Last modified by alozano (2414) Numerical id 8 Author alozano (2414) Entry type Definition Classification msc 11G07 Classification msc 11G05 Classification msc 14H52 Related topic CanonicalHeightOnAnEllipticCurve Related topic BirchAndSwinnertonDyerConjecture Related topic Regulator Defines elliptic regulator Defines height matrix