---
date: 2022-02-15 12:12
modification date: Tuesday 15th February 2022 12:12:05
title: Neron-Tate height
aliases: [Neron-Tate height, canonical height, elliptic regulator, "Neron-Tate height pairing", "Neron-Tate", "Néron-Tate height"]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
- Refs:
	- [Height](Height.md) functions in number theory: 
	  <https://people.math.harvard.edu/~sli/notes/math383_w18_notes.pdf#page=1> #resources/course-notes 
- Links:
	- [Mordell-Weil](Unsorted/Mordell-Weil.md)
	- [abelian variety](abelian%20variety)
	- [global field](Unsorted/global%20field.md)
	- [elliptic curves](MOCs/elliptic%20curve.md)


---

# Neron-Tate height

A [quadratic form](quadratic%20form.md) on the [Mordell-Weil](Mordell-Weil.md) group of [rational points](rational%20points.md) on an [elliptic curve](elliptic%20curve.md), or more generally an [abelian variety](Unsorted/abelian%20variety.md).

![](attachments/Pasted%20image%2020220502095053.png)

# Elliptic regulator

The bilinear form associated to the canonical height $\hat{h}$ on an elliptic curve $E$ is
\[
\langle P, Q\rangle=\frac{1}{2}(\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q)) .
\]
The elliptic regulator of $E / K$ is
$$
\Reg(E\slice K)\da \det G,\qquad G_{ij} \da \inp{P_i}{P_j}
$$
where $\ts{P_i}$ is a basis of $E(K)\tensor_\ZZ \QQ$, the free part of the [Mordell-Weil group](Mordell-Weil%20group) of $E$.