TITLE: Division by 3 on elliptic curve
QUESTION [10 upvotes]: There is a classical theorem (cf. Theorem 4.1 on Husemoller "Elliptic Curve" book) that states conditions on the coordinates of a point P in an elliptic curve to be twice another point. The result is the following:
Theorem: Let $E$ be an elliptic curve defined over a field $K$ by the equation
$$
E:y^2=(x-\alpha)(x-\beta)(x-\gamma),\quad\mbox{with $\alpha,\beta,\gamma\in K$}. 
$$
For $(x',y')\in E(K)$ there exists $(x,y)\in E(K)$ with $2(x,y)=(x',y')$ if and only if $x'-\alpha$, $x'-\beta$ and $x'-\gamma$ are squares.
I would like to know if there is something similar for 3. That is, condition on the coordinates $(x',y')\in E(K)$ such that there exists $(x,y)\in E(K)$ with $3(x,y)=(x',y')$.

REPLY [8 votes]: In this paper by Ed Schaefer and myself, there is a reasonably explicit description of `3-descent' on an elliptic curve. There is a somewhat more detailed version of the paper here.
Assuming that your curve $E$ over $K$ is such that the Galois action on the points of order 3 is transitive, you look at the field $L = K(P_0)$, where $P_0$ is any point of order 3 on $E$. Then $P_0 = (\xi,\eta) \in E(K)$ and you can consider the tangent line $y = \lambda x + \mu$ at this point $P_0$. The map
$$E(K) \longrightarrow L^\times/L^{\times 3}, \qquad P \longmapsto f(P) = y(P) - \lambda x(P) - \mu$$
(modulo $L^{\times 3}$, where $L^{\times 3}$ means the subgroup of cubes) has kernel $3 E(K)$. This means that $P$ is divisible by 3 in $E(K)$ if and only if ($P = O$ or) $f(P)$ is a cube in $L$.
This is in fact the correct generalization of Joe's reply (for $m=3$) to the case when you don't have full rational 3-torsion: $f$ is the correctly scaled function with divisor $3(P_0) - 3(O)$, and working over $L$ basically means that we look at all the $f$'s for the various choices of $P_0$ at the same time.
When the Galois action is not transitive, you may have to consider two such functions $f$ (such that the orbits of the corresponding points generate $E[3]$ and have total cardinality not a multiple of 3).