Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 125 / 1 / c

Equality of the two point groups implies equality of their orders. Since ${\alpha }^2+{\beta }^2=a^2-2p$,soIts two integral solutions are $a=p+1$ and $a=-p$. The Hasse theorem for elliptic curves excludes the first for every prime and permits the second only when $p\le 4$. Thus $p=2$ or $p=3$.

Both occur. Over ${\mathbb{F}}_2$, the smooth curve $y^2+y=x^3+x$ has five rational points and trace $-2$. Over ${\mathbb{F}}_3$, the smooth curve $y^2=x^3-x+1$ has seven rational points and trace $-3$. In either case the point-count formula gives $\#E({\mathbb{F}}_{p^2})=\#E({\mathbb{F}}_p)$. Since $E({\mathbb{F}}_p)\subseteq E({\mathbb{F}}_{p^2})$, equal orders give equality of groups.

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