Neron-Tate height

A quadratic form on the Mordell-Weil group of rational points on an elliptic curve, or more generally an abelian variety.

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Elliptic regulator

The bilinear form associated to the canonical height \(\widehat{h}\) on an elliptic curve \(E\) is \begin{align*} \langle P, Q\rangle=\frac{1}{2}(\widehat{h}(P+Q)-\widehat{h}(P)-\widehat{h}(Q)) . \end{align*} The elliptic regulator of \(E / K\) is \begin{align*} \operatorname{Reg}(E_{/ {K}} )\coloneqq\operatorname{det}G,\qquad G_{ij} \coloneqq{\left\langle {P_i},~{P_j} \right\rangle} \end{align*} where \(\left\{{P_i}\right\}\) is a basis of \(E(K)\otimes_{\mathbf{Z}}{\mathbf{Q}}\), the free part of the Mordell-Weil group of \(E\).

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Mordell-Weil group in Quick_Notes/2021-04-15, -Unsorted/Mordell-Weil, -Unsorted/Neron-Tate height
height

Unsorted/Neron-Tate height
Birch and Swinnerton-Dyer conjecture

On the free abelian group \(E(K) / E(K)_{\text {tor }}\), there is areal-valued positive definite bilinear form, the Néron-Tate height pairing. From this one can construct a regulator Reg \({ }_{E} / K\), which is the determinant of the matrix of this bilinear form with respect to a basis.