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Viewing Version
4
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'regulator of an elliptic curve'
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| Title of object: |
regulator of an elliptic curve |
| Canonical Name: |
RegulatorOfAnEllipticCurve |
| Type: |
Definition |
| Created on: |
2006-11-08 21:02:20 |
| Modified on: |
2006-11-09 14:20:17 |
| Classification: |
msc:14H52, msc:11G05, msc:11G07 |
| Defines: |
elliptic regulator, height matrix |
Revision comment (for changes between this and next version):
| Changes for correction #11623 ('clarify'). |
Preamble:
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Content:
Let $E/\Rats$ be an elliptic curve and let $\langle \cdot, \cdot \rangle$ be the N\'eron-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 $E/\Rats$.
\begin{defn}
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 {\bf 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 $H=1$.
\end{defn}
\begin{defn}
The {\bf \PMlinkescapetext{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.
\end{defn}
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 N\'eron-Tate pairing. |
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