elliptic curves over finite fields

**zverik136** · November 29th 2009, 11:59 AM

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

**tonio** · November 30th 2009, 03:44 AM

Originally Posted by zverik136

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 $p=2\!\!\!\!\pmod 3$ , the map $u\rightarrow u^3$ is an automorphism of $\mathbb{F}_p^{*}$ , and thus $u\rightarrow \left((u^2-B)^{1\slash 3},u)\right)$ is a bijection between $\mathbb{F}_p\,\mbox{ and } E_p=$ the set of points of the elliptic curve in $\mathbb{F}_p^2$ . Now add the point at infinity in projective space and we're done.

Tonio

**zverik136** · November 30th 2009, 07:21 AM

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".

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