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height function
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(Definition)
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Definition 1 Let $A$ be an abelian group. A height function on $A$ is a function $h\colon A\to \mathbb{R}$ with the properties:
- For all $Q\in A$ there exists a constant $C_1$ , depending on $A$ and $Q$ , such that for all $P\in A$ : $$ h(P+Q)\leq 2h(P) +C_1$$
- There exists an integer $m \geq 2$ and a constant $C_2$ , depending on $A$ , such that for all $P\in A$ : $$ h(mP)\geq m^2h(P) -C_2$$
- For all $C_3 \in \mathbb{R}$ , the following set is finite: $$\{ P\in A: h(P)\leq C_3\} $$
Examples:
- For $t=p/q \in \mathbb{Q}$ , a fraction in lower terms, define $H(t)=\max \{\mid p \mid ,\mid q \mid \}$ . Even though this is not a height function as defined above, this is the prototype of what a height function should look like.
- Let $E$ be an elliptic curve over $\mathbb{Q}$ . The function on $E(\mathbb{Q})$ , the points in $E$ with coordinates in $\mathbb{Q}$ , $h_x\colon E(\mathbb{Q})\to \mathbb{R}$ : $$h_x(P)={{\log H(x(P)),\quad if\ P\neq 0} \brace {0,\quad if\ P=0}}$$ is a height function ($H$ is defined as above). Notice that this depends on the chosen Weierstrass model of the
curve.
- The canonical height of $E/\mathbb{Q}$ (due to Neron and Tate) is defined by: $$h_C(P)=1/2 \lim_{N\to \infty} 4^{(-N)}h_x([2^N]P)$$ where $h_x$ is defined as in (2).
Finally we mention the fundamental theorem of ``descent'', which highlights the importance of the height functions:
Theorem 1 (Descent) Let $A$ be an abelian group and let $h\colon A \to \mathbb{R}$ be a height function. Suppose that for the integer $m$ , as in property (2) of height, the quotient group $A/mA$ is finite. Then $A$ is finitely generated.
- 1
- Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
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"height function" is owned by alozano.
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Cross-references: finitely generated, quotient group, height, theorem, curve, Weierstrass model, coordinates, points, elliptic curve, even, terms, fraction, finite, integer, properties, function, abelian group
There are 5 references to this entry.
This is version 2 of height function, born on 2003-08-04, modified 2003-08-05.
Object id is 4549, canonical name is HeightFunction.
Accessed 7194 times total.
Classification:
| AMS MSC: | 14H52 (Algebraic geometry :: Curves :: Elliptic curves) |
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Pending Errata and Addenda
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