|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
regulator of an elliptic curve
|
(Definition)
|
|
|
Let $E/\Rats$ be an elliptic curve, let $E(\Rats)$ 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 $E/\Rats$ .
Definition 1 Let $E/\Rats$ be an elliptic curve and let $\{P_1,\ldots,P_r\}$ be a set of generators of the free part of $E(\Rats)$ , i.e. the points $P_i$ generate $E(\Rats)$ modulo the torsion subgroup $E_{\operatorname{tors}}(\Rats)$ . The height matrix of $E/\Rats$ is the $r\times r$ matrix $H$ whose $ij$ 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 regulator of $E/\Rats$ (or the elliptic regulator), denoted by $\operatorname{Reg}(E/\Rats)$ or $R_{E/\Rats}$ is defined by $$\operatorname{Reg}(E/\Rats)=\det(H)$$ 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 $E(\Rats)$ modulo torsion, with respect to the quadratic form defined by the Néron-Tate pairing.
|
"regulator of an elliptic curve" is owned by alozano.
|
|
(view preamble | get metadata)
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 2651 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) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
