Solving $P+P=Q$ on the curve $y^2=x^3-n^2x$

Question

If $P=(x_0,y_0)$ is a rational point on the curve $y^2=x^3-n^2x$, let $Q=2P=P+P=(x_1,y_1)$. Then $x_1$, $x_1+n$ and $x_1-n$ are all rational squares (see for example Ch 1 of the book Elliptic curves and cryptography by L. Washington). I would like to know what can be said instead by going in the reverse direction, that is, suppose we have a rational point $Q=(x_1,y_1)$ on the curve, and we solve the equations corresponding to the group law addition to get a point $P=(x_0,y_0)$ such that $P+P=Q$. I did this and found $x_0=x_1\pm \sqrt{x_1^2-n^2}\pm \sqrt{2x_1^2\pm 2x_1 \sqrt{x_1^2-n^2}}$, where using $+++,+-+,-+-,---$ for the three $\pm$ signs actually gives four solutions. Clearly $x_0$ doesn't have to be rational, but what extension of $\mathbb Q$ do we need? For a specific example when $n=5$, consider the point $(x_1,y_1)=(-4,6)$ on the curve $y^2=x^3-25x$. Notice that $-4=(2i)^2$, $-4+5=1^2$ and $-4-5=(3i)^2$, and using the $+++$ sign pattern in the above formula we get $(x_0,y_0)=(2+i,1-7i)$. So $x_0$ is in the extension of $\mathbb Q$ obtained by simply adjoining $\sqrt{x_1}=\sqrt{-4}$. Is this always the case? I guess my question could also be phrased: what extension of a field $k$ do we need to halve points of an elliptic curve defined over $k$? I imagine this is all well-known and written somewhere, but I am new to elliptic curves. I searched the archives but I did not find an answer. I may not know the right search words to use.

Answer 1

Note that this curve is related to the congruent number problem. More precisely, in Proposition 9.19 in Introduction to Elliptic curves and Modular Forms by N. Koblitz, it is stated that there is a 1-1 correspondence between pair of points $(x,\pm y) \in 2E_{n}(\mathbb{Q})-O$ and right-angled triangles with rational sides $X,Y,Z$ and area $n$. One side of this correspondence is given by \begin{equation} (x, \pm y) \mapsto (X,Y,Z)=(\sqrt{x+n} - \sqrt{x-n},\sqrt{x+n}+\sqrt{x-n},2\sqrt{x} ). \end{equation}

So, a point $P=(x,y)$ can be "halved over $\mathbb{Q}$ if $\sqrt{x},\sqrt{x-n},\sqrt{x+n} \in \mathbb{Q}$.

Answer 2

This is a partial answer, but I do not have enough reputation to comment.

Note that if $m$ is the order of $Q$ and $m$ is not even, $P = [d]Q$, where $d = 2^{-1} \pmod m$.

Thus you can only limit yourself to studying the points $Q$ of even order.