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 $⟨\cdot ,\cdot ⟩$ be the Néron-Tate pairing:

$$⟨P,Q⟩=\widehat{h}(P+Q)-\widehat{h}(P)-\widehat{h}(Q)$$

where $\widehat{h}$ is the canonical height on the elliptic curve $E/\mathbb{Q}$.

Definition 1.

Let $E\mathrm{/}\mathrm{Q}$ be an elliptic curve and let $\mathrm{\{}{P}_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{P}_r\mathrm{\}}$ be a set of generators of the free part of $E\mathit{}\mathrm{(}\mathrm{Q}\mathrm{)}$, i.e. the points $P_i$ generate $E\mathit{}\mathrm{(}\mathrm{Q}\mathrm{)}$ modulo the torsion subgroup $E_{\mathrm{tors}}\mathit{}\mathrm{(}\mathrm{Q}\mathrm{)}$. The height matrix of $E\mathrm{/}\mathrm{Q}$ is the $r\mathrm{\times }r$ matrix $H$ whose $i\mathit{}j$th component is $\mathrm{⟨}{P}_i\mathrm{,}{P}_j\mathrm{⟩}$, i.e.

$$H=(⟨P_i,P_j⟩).$$

If $r\mathrm{=}\mathrm{0}$ then we define $H\mathrm{=}\mathrm{1}$.

Definition 2.

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

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

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