PlanetMath: regulator of an elliptic curve

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regulator of an elliptic curve

(Definition)

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:

$\displaystyle \langle P,Q \rangle=\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q)$

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 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 whose th component is $\langle P_i, P_j \rangle$ , i.e.

$\displaystyle H = (\langle P_i, P_j \rangle).$

If then we define .

Definition 2 The regulator of $E/\mathbb{Q}$ (or the elliptic regulator), denoted by $\operatorname{Reg}(E/\mathbb{Q})$ or $R_{E/\mathbb{Q}}$ is defined by

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

where 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.

"regulator of an elliptic curve" is owned by alozano.

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Also defines:

elliptic regulator, height matrix

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Cross-references: quadratic form, torsion, domain, volume, regulator of a number field, similarities, component, matrix, torsion subgroup, generate, generators, canonical height, pairing, curve, points, rational, group, elliptic curve
There are 2 references to this entry.

This is version 5 of regulator of an elliptic curve, born on 2006-11-08, modified 2007-04-09.
Object id is 8535, canonical name is RegulatorOfAnEllipticCurve.
Accessed 1436 times total.

Classification:

AMS MSC:	14H52 (Algebraic geometry :: Curves :: Elliptic curves)
	11G05 (Number theory :: Arithmetic algebraic geometry :: Elliptic curves over global fields)
	11G07 (Number theory :: Arithmetic algebraic geometry :: Elliptic curves over local fields)

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