Thread: 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.

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 has order 2, then and therefore which has no non-zero solution in

so, there is no point of order 2 on the curve. if has order at least 3, then which gives us using these possible values of find possible values of by solving

finally accept only those which have finite order. (remember the theorem only gives a necessary condition!)