regulator of an elliptic curve

Let E/ be an elliptic curveMathworldPlanetmath, let E() be the group of rational points on the curve and let , be the Néron-Tate pairing:


where h^ is the canonical heightPlanetmathPlanetmath on the elliptic curve E/.

Definition 1.

Let E/Q be an elliptic curve and let {P1,,Pr} be a set of generatorsPlanetmathPlanetmathPlanetmath of the free part of E(Q), i.e. the points Pi generate E(Q) modulo the torsion subgroup Etors(Q). The height matrix of E/Q is the r×r matrix H whose ijth component is Pi,Pj, i.e.


If r=0 then we define H=1.

Definition 2.

The of E/Q (or the elliptic regulator), denoted by Reg(E/Q) or RE/Q is defined by


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() modulo torsionPlanetmathPlanetmath, with respect to the quadratic formMathworldPlanetmath 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