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Math Help - elliptic curves: points of finite order

  1. #1
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    Question elliptic curves: points of finite order

    I was wondering if anyone could explain the method of finding all the points of finite order over C(\mathbb{Q}), with C: y^2=x^3 + px, where p is a prime bigger or equal to 2.

    this would explain a lot of the theory to me.
    also - do worked examples or solutions to exercises from cassels, or silverman (+tate) exist? missing the ability to apply things i have heard in lectures

    thank you.
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  2. #2
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    Quote Originally Posted by zverik136 View Post
    I was wondering if anyone could explain the method of finding all the points of finite order over C(\mathbb{Q}), with C: y^2=x^3 + px, where p is a prime bigger or equal to 2.

    this would explain a lot of the theory to me.
    also - do worked examples or solutions to exercises from cassels, or silverman (+tate) exist? missing the ability to apply things i have heard in lectures

    thank you.
    apply Nagell-Lutz theorem: first of all the (non-zero) torsion points have integer coordinates. if P(a,b) has order 2, then b=0 and therefore a^3+pa=0, which has no non-zero solution in \mathbb{Z}.

    so, there is no point of order 2 on the curve. if P(a,b) has order at least 3, then b^2 \mid 4p^3, which gives us b=\pm 2, \ \pm p, \ \pm 2p. using these possible values of b find possible values of a by solving

    b^2=a^3+pa. finally accept only those P(a,b) which have finite order. (remember the theorem only gives a necessary condition!)
    Last edited by NonCommAlg; October 26th 2009 at 02:46 AM.
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