Are there period points on an elliptic curve?

Question

There are a variety of elliptic curves which are often used in cryptography for things like key exchanges. It seems it would be problematic if points on the elliptic curve can be periodic. What I mean is given starting point on the curve $P = (x,y)$. Can it be that there is some m such that $mP = P$? If there are, what are the requirements to find such a curve? I'm curious as to the behavior of points on a variety of elliptic curve and if you can find periodic points (and if there is a formula for finding such a thing).

Answer 1

If $mP=P$, then $(m-1)P = \mathcal{O}$, so $P$ is a torsion point of order dividing $m-1$. Depending on the field of definition of your elliptic curve, you can find torsion points of various orders. For example, over $\mathbb{Q}$, you can find torsion points of order $m-1=1,2,3,4,5,6,7,8,9,10$, and $12$.