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

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- Nov 29th 2009, 11:59 AMzverik136elliptic curves over finite fields
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 - Nov 30th 2009, 03:44 AMtonio

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 - Nov 30th 2009, 07:21 AMzverik136
thank you! i did mean over projective space, sorry.

how about for p= 3 mod 4, with Y^2=X^2(X+A), defined over F_p. what bijection could be found? the hint that cassels gives is "consider +/-X together".