# elliptic curves: points of finite order

• October 26th 2009, 12:24 AM
zverik136
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.
• October 26th 2009, 01:22 AM
NonCommAlg
Quote:

Originally Posted by zverik136
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!)