Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 125 / 2 / c / Solution

A positive integer $D$ is a congruent number when it is the area of a right triangle with positive rational side lengths. For a point $P=(x,y)$ with $y\ne 0$ onthe formulasgive $a^2+b^2=c^2$ and $ab/2=D$, after changing signs if necessary. Conversely, a rational right triangle of area $D$ givesso these constructions are inverse up to the usual sign choices.

It remains to distinguish torsion. If an odd prime $\ell$ divided the order of a rational torsion point, choose by the Dirichlet theorem on primes in arithmetic progressions a good prime $p\equiv 3(\mathrm{mod}4)$ for which $p\not\equiv -1(\mathrm{mod}\ell )$. Part (a) gives $\#E_D({\mathbb{F}}_p)=p+1$, while part (b), applied to the formal group of an elliptic curve, makes reduction injective on $\ell$-power torsion because $\ell$ is a unit in ${\mathbb{Z}}_p$. This is impossible. Similarly, a good prime $p\equiv 3(\mathrm{mod}8)$ shows that the rational $2$-primary torsion has order at most four. Sincealready form the full rational 2-torsion,The triangle construction uses exactly the points with $y\ne 0$, which are therefore nontorsion. By the Mordell-Weil theorem, such a point exists exactly when the free part has positive rank. Hence