Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 125 / 5

For $P=[x_0:\cdots :x_N]\in {\mathbb{P}}^N(\mathbb{Q})$, choose coprime integer coordinates and define the projective heightWrite the morphism $F:{\mathbb{P}}^1\to {\mathbb{P}}^1$ as $F=[F_0:F_1]$, where $F_0,F_1\in \mathbb{Z}[X,Y]$ are homogeneous of degree $d$ with no common zero. Bounding each polynomial by the sum of the absolute values of its coefficients gives

Because $F_0$ and $F_1$ have no common projective zero, the Projective Nullstellensatz gives an integer $m\ge d$ and homogeneous polynomials $A_{ij}$ such that suitable nonzero integer multiples of $X^m$ and $Y^m$ lie in the ideal $(F_0,F_1)$. Evaluating at primitive coordinates and using the same coefficient bound givesafter absorbing the bounded common divisor of $F_0(x,y)$ and $F_1(x,y)$ into $C$. ThereforeThis is height growth under a morphism of the projective line.

Let $x:E\to {\mathbb{P}}^1$ be the $x$-coordinate map and define the naive logarithmic height byThe duplication formula induces a degree-four morphism on the $x$-line, so part (a) givesuniformly in $P$. The telescoping sequence $4^{-n}h(2^{n}P)$ is therefore Cauchy. Define the canonical height of an elliptic curve bySumming the geometric error series proves that $|h(P)-\hat{h}(P)|$ is bounded. If another function has this bounded-difference property and scales by four under doubling, evaluating the bounded difference at $2^{n}P$ and dividing by $4^n$ proves uniqueness.

The addition law and part (a) give the approximate parallelogram identityReplace $P,Q$ by $2^{n}P,2^{n}Q$, divide by $4^n$, and let $n\to \infty$. The error disappears, leaving the exact parallelogram lawTogether with $\hat{h}(2P)=4\hat{h}(P)$, induction on $|m|$ yieldsfor every integer $m$.

Let $T$ have order $m>0$. Property (ii) gives $\hat{h}(T)=0$, sinceIt also givesThereforefor every rational torsion point of an elliptic curve. The construction in part (i), quadraticity in part (ii), and this identity establish existence; the bounded-difference and doubling argument in part (i) establishes uniqueness.

Since $\hat{h}-h$ is bounded, a bound on $\hat{h}(P)$ bounds the projective height of $x(P)$. The Northcott theorem gives only finitely many possible rational $x$-coordinates, and each has at most two points above it. Hence

By the Mordell-Weil theorem, $E(\mathbb{Q})\cong {\mathbb{Z}}^r\oplus E(\mathbb{Q}{)}_{\mathrm{tors}}$. The canonical height is a positive-definite quadratic form on the lattice ${\mathbb{Z}}^r$, so canonical-height lattice-point growth givesIf $r=0$, this count is bounded; if $r=1$, it is $O(\sqrt{B})$. Consequently