An elliptic curve is defined over the field of real numbers:


$y^2=x^3 + ax + b$


A point P and scalar n can be multiplied using a combination of [point doubling and adding](http://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication#Point_multiplication).


What about point division? Given point P on the curve, what procedures exist for computing point P/n?


This question usually comes up in the context of finite fields, but I'm curious about the field of real numbers. Is division even possible in this case, and if so, under what conditions?



Is elliptic curve point division defined over the field of real numbers?
The multiplication by $n$ map is surjective on the set of complex points, so you can always divide there, and a point has $n^2$ complex inverse images. Over the reals, $E(\mathbb{R})$ is isomorphic, as a real Lie group, to either the circle group $S^1$ or
to two copies $S^1\times\mathbb{Z}/2\mathbb{Z}$. In the former case, every real point has a exactly $n$ real $n$'th roots, in the latter it depends on whether $n$ is odd or even and on whether the point is on the identity component or not. One can easily compute these roots using the standard formulas that parametrize $E(\mathbb{C})$, or even better,
by an isomorphism $E(\mathbb{C})\cong\mathbb{C}^\*/q^{\mathbb{Z}}$ with $q$ real. See for example Chapter V Section 2 of my book "Advanced Topics in the Arithmetic of Elliptic Curves," but of course the theory has been well known for a very long time. (Felipe is probably right, this question would be better suited for MathStackexchange.)


