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I have been reading through Silverman's classic text on elliptic curves and I just can't seem to wrap my head around the height functions. It just kind of shows up. What exactly does the height describe? Why is it important (for instance why do we want to compute the canonical height). Wikipedia says it measures the complexity of a point, but that seems a little cryptic to me.

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Intuitively it is a crude measure of how "complicated" the point is - a point with small height is really a point with small numerator and denominator. In fancier terms it would be an adelic metric on the hyperplane section. – user27126 Apr 10 '13 at 14:43
Would you mind writing an example of the "crude" measure as an answer? I'm not familiar with hyperplanes either so any insight is appreciated. – Brent J Apr 11 '13 at 1:49

The Mordell-Weil Theorem says that if $E$ is an elliptic curve over a number field $K$, then $E(K)$ is a finitely generated abelian group. I know the canonical height is useful to prove this theorem. Let me draw an outline of the strategy, in the case $K=\mathbb Q$.

Fact. If $M$ is a finitely generated abelian group, and $n\geq 1$ is an integer, then $M/nM$ is finite.

Unfortunately, the converse is false (e.g. $\mathbb Q/n\mathbb Q$ is finite but $M=\mathbb Q$ is not finitely generated). I claim that the canonical height is what we need to get a converse of the Fact.

Indeed, assume we know that $E(\mathbb Q)/2E(\mathbb Q)$ is finite. Then, this fact together with the theory of heights give us Mordell-Weil, in a way that I now try to explain.

There is a naive notion of "height" of a rational number: if $a/b\in\mathbb Q$ is a reduced fraction, we can define $H:\mathbb P^1(\mathbb Q)\to \mathbb Q$ by $H(a:b)=\max\,\{|a|,|b|\}$ (we may call this the height of $a/b$), and also $h:\mathbb P^1(\mathbb Q)\to \mathbb R$ by $h(a:b)=\log\,H(a:b)$. One builds the canonical height on $E$ \begin{equation} \hat h:E(\mathbb Q)\longrightarrow\mathbb R \end{equation} starting from $h$. This function $\hat h$ is important because it has the following properties. It is a quadratic form on $E(\mathbb Q)$ such that:

  1. $\hat h(P)=0$ if and only if $P$ is a torsion point;
  2. For every $c>0$, the set $S_c:=\{P\in E(\mathbb Q)\,|\,\hat h(P)\leq c\}$ is finite.

With the data described so far, one can prove Mordell-Weil. Indeed, recall that we are assuming $E(\mathbb Q)/2E(\mathbb Q)$ is finite. With this assumption, it is easy to prove the following:

Assertion. Let $c>0$ be such that $S_c$ contains a set of representatives of $E(\mathbb Q)$ modulo $2E(\mathbb Q)$. Then $S_c$ generates $E(\mathbb Q)$. (And $S_c$ is finite by 2!)

[Note that such a $c$ exists: take $P_1,\dots,P_s\in E(\mathbb Q)$ to be representatives of $E(\mathbb Q)$ modulo $2E(\mathbb Q)$, and let $c:=\max\,\{\hat h(P_1),\dots,\hat h(P_s)\}$.

Of course, the hard part is to show that $E(\mathbb Q)/2E(\mathbb Q)$ is finite. In fact, one shows that for every $n\geq 1$, \begin{equation} E(\mathbb Q)/nE(\mathbb Q)\subset \textrm{Sel}^n(E/\mathbb Q), \end{equation} where $\textrm{Sel}^n(E/\mathbb Q)$ is a finite group.]

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Thanks! That certainly answers my question of why it is important! – Brent J Apr 11 '13 at 1:47

This is a bit too long to put in comment, so I'm putting it here.

There are numerous counting problems in Mathematics. Some examples would be:

  • How many primes are there in natural numbers/ring of integers in a number field?
  • How many elliptic curves are there?

In each question there are "many" objects you want to count, and it's natural to ask about the density of these objects, e.g. The density problem in the first question is the prime number theorem. But for density to make sense, we need to specify enumerate the objects so that we can write down an expression for density.

One way to enumerate objects would be to introduce the notion of size. In the case of primes in natural numbers, it would be the size of the number so that we would look at the limit $$\lim_{x \to \infty} \frac{\text{number of primes less than $x$}}{\text{number of positive integers less than $x$}} $$ as the density of primes. To count primes in the ring of integers, one would use the notion of norm of ideal as the size of ideal, and we can consider $$\lim_{x \to \infty} \frac{\text{number of prime ideals with norm less than $x$}}{\text{number of integral ideals with norm less than $x$}} $$ as the density. Similarly for counting elliptic curves, one may use conductor or discriminant to measure the "size" of an elliptic curve.

Now there are many rational points on elliptic curves. (Mordell-Weil) It's then natural to ask about the density of rational points on elliptic curves, and to do this, we would need the notion of the size of a rational point. Very naively for $\mathbb{P}^n(\mathbb{Q})$, one can impose a notion of size: The size of $[a_1,\cdots,a_n]$ with $a_1,\cdots,a_n \in \mathbb{Z}$ and gcd being 1 would have size $h([a_1,\cdots,a_n]) = \max\{|a_1|,\cdots,|a_n|\}$. Examples in $\mathbb{P}^1$ include

$$h([2,1]) = 2, h([343, 512]) = 512$$

If you embed $\mathbb{A}^1$ into $\mathbb{P}^1$ by $t \to [t,1]$, you can see that the two examples are saying that height of $2 = 2/1$ is 2, and height of $343/512$ is 512. The latter fraction does look more complicated, and hopefully you would agree that $343/512$ should have a larger "size" in an appropriate sense.

Of course, this notion of size is not dependent on the counting problem; the counting problem is just one situation where it comes up naturally. In fact you even use the notion of size in the proof of Mordell-Weil, as explained in the other (excellent) answer. Equipped with the notion of size, and weak Mordell-Weil ($E(\mathbb{Q})/2E(\mathbb{Q})$ is finite), one can prove Mordell-Weil quite naturally, as is done in Chapter 8.3 of Silverman.

The naive height would behave a bit like a bilinear form, but up to an error. (Theorem 8.6.2 in Silverman) In the case of elliptic curves/abelian varieties there is a notion of canonical height (Neron-Tate height) where the height is really a bilinear form, so is a better notion of height. Fun fact: This can be reinterpreted in Arakelov geometry as some intersection pairing, so height has some geometric meaning as well.

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