How would one show that the number of points on the elliptic curve y^2=x^3+B, defined over F_p is p+1, when p is congruent to 2 mod 3?
Thank you
You mean over projective space, right? Anyway, as $\displaystyle p=2\!\!\!\!\pmod 3$ , the map $\displaystyle u\rightarrow u^3$ is an automorphism of $\displaystyle \mathbb{F}_p^{*}$, and thus $\displaystyle u\rightarrow \left((u^2-B)^{1\slash 3},u)\right)$ is a bijection between $\displaystyle \mathbb{F}_p\,\mbox{ and } E_p=$ the set of points of the elliptic curve in $\displaystyle \mathbb{F}_p^2$. Now add the point at infinity in projective space and we're done.
Tonio