Finding elliptic curve with exactly p+1 number points over F_p

Question

Hi all I am beginner in Elliptic Curves. I want to design an elliptic curve with exactly $p+1$ points over $\mathbb{F}_p$. Any approach towards starting to solve this problem or recent progress or any references would be really helpful.

Thanks

Answer 1

The quantity $1+p - |E(\mathbb F_p)|$ is usually denoted $a_p$, and so you are asking for curves for which $a_p = 0$. When $p \geq 5$, this condition is equivalent to the elliptic curve being supersingular.

In general, there is a finite positive number of elliptic curves over $\overline{\mathbb F}_p$, and they can always be defined over $\mathbb F_{p^2}$ (i.e. their $j$-invariants necessarily lie in this field). You are looking for supersingular $j$-invariants that actually lie in $\mathbb F_p$.

One way to find such curves is to take curves with CM by the ring of integers in $\mathbb Q(\sqrt{-p})$, which are defined over the Hilbert class field of $\mathbb Q(\sqrt{-p})$, and then reduce them modulo a prime lying over $\sqrt{-p}$ (perhaps after making an extension totally ramified over this prime in order to obtain good reduction). More details of this construction are given in section 3 of this paper by Andrew Baker.