Results 1 to 3 of 3

Math Help - elliptic curves over finite fields

  1. #1
    Newbie
    Joined
    Oct 2009
    Posts
    11

    elliptic 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
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by zverik136 View Post
    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
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2009
    Posts
    11
    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".
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Distinct Bases for Finite Vector Spaces over Finite Fields
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: May 29th 2011, 01:31 PM
  2. Moduli space for elliptic curves
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: June 8th 2010, 04:04 PM
  3. elliptic curves: interesting question
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: October 26th 2009, 02:55 AM
  4. elliptic curves: points of finite order
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: October 26th 2009, 01:22 AM
  5. Elliptic curves
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: November 19th 2008, 09:25 PM

Search Tags


/mathhelpforum @mathhelpforum