regulator of an elliptic curve
Let be an elliptic curve, let be the group of rational points on the curve and let be the Néron-Tate pairing:
where is the canonical height on the elliptic curve .
Definition 1.
Let be an elliptic curve and let be a set of generators of the free part of , i.e. the points generate modulo the torsion subgroup . The height matrix of is the matrix whose th component is , i.e.
If then we define .
Definition 2.
The of (or the elliptic regulator), denoted by or is defined by
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 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 |