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 $i j$ 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