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

If $p\mid D$, square-freeness gives the cuspidal reduction $y^2=x^3$. Its nonsingular group is isomorphic to the additive group ${\mathbb{F}}_p$, so it is cyclic of order $p$.

Suppose $p\nmid D$. The reduction is an elliptic curve. Since $p\equiv 3(\mathrm{mod}4)$, $-1$ is a quadratic nonresidue. Pairing $x$ with $-x$ in the quadratic-character sum and usingshows that the two contributions cancel. Therefore $\#\tilde{E}({\mathbb{F}}_p)=p+1$. The three roots $0,D,-D$ are distinct, so the full 2-torsion is rational. A cyclic group has at most two elements killed by $2$, hence $\tilde{E}({\mathbb{F}}_p)$ is noncyclic. Thus

A one-dimensional commutative formal group law over a ring $R$ is a series $F(X,Y)\in R[[X,Y]]$ satisfyingAn isomorphism $u:F\to G$ is a series $u(T)\in TR[[T]]$ with a compositional inverse and

The multiplication series is defined recursively by $[0{]}_F=0$, $[1{]}_F=T$ and $[n+1{]}_F=F([n{]}_F,T)$, with the formal inverse handling negative $n$. Its linear term isBy the invertible morphism criterion for formal group laws, it is an isomorphism exactly when its linear coefficient $n$ is a unit of $R$. Indeed, when $n\in R^{\times }$, recursive coefficient comparison constructs a unique compositional inverse; applying the morphism identity for $[n{]}_F$ shows that the inverse also respects $F$. Conversely, an invertible series must have a unit linear coefficient. Therefore

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