Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 125 / 1 / c

Let $\pi$ be the Frobenius isogeny and let $a=p+1-N$ be its Trace of Frobenius. Its characteristic equation is ${\pi }^2-[a]\pi +[p]=0$, so the trace of the square of an elliptic-curve endomorphism is $a^2-2p$. The elliptic-curve point count over a finite field therefore givesThe prime-square point-count formula is consequentlyOne can also obtain it directly from the degree of an isogeny: the maps $1-\pi$ and $1+\pi$ are separable isogenies, with degrees $p+1-a$ and $p+1+a$. Their composition is $1-{\pi }^2$, whose kernel of an isogeny is exactly $E({\mathbb{F}}_{p^2})$. Multiplicativity of degree gives the same product.