There is a question in Hartshorne which asks for the group law on the elliptic curve defined by the equation $$y^2+y= x^3-x,$$ namely to find an expression for $P+Q$, where $P = (0,0)$ and $Q=(a,b)$ in terms of $a$ and $b$. I am aware of the explicit formulas for the group law on elliptic curves (for example, as proven in chapter 3 of Silverman's text), but these explicit formulas (to my knowledge) are never mentioned in Hartshorne.

I am looking for a way of deducing the explicit formulas purely from the definition of the group law in terms of divisors. Explicitly $P+Q = R$ if and only if $P+Q\sim R+P_0$ (which is obviously equivalent, yet "coordinate-free" so to speak).


Edited. (Thanks to Ravi for spotting the original error!)

As Ravi pointed out, the embedding of the abstract elliptic curve in $\mathbb P^2$ can be described as follows: one picks a point $P_0$ on the curve, and embedds using the complete linear system $|3P_0|$. Once embedded in $\mathbb P^2$, the divisor $3P_0$ is then equivalent to the class of the hyperplane $H$ associated with this embedding in $\mathbb P^2$.

In your case, I believe that $P_0 = [0 : 1 : 0]$ is the point that Hartshorne has in mind for this purpose. The fact that $3P_0 \sim H$ is clear from the fact that the hyperplane $z = 0$ has triple intersection with the elliptic curve at $P_0$.

So having identified $3P_0 \sim H$, one can use the following statement to characterise the group law: $$ P + Q + R = 0 \iff P + Q + R \sim H $$

For the purposes of your exercise from Hartshorne, I would compute $P + Q$ in two stages. First, I would determine $S := - (P + Q)$ by finding the hyperplane that intersects the elliptic curve at $P$ and $Q$ and taking $S$ to be the third point of intersection. Second, I would determine $R := - S = - ( P_0 + S)$ by finding the third intersection point for the hyperplane that intersects the elliptic curve at $S$ and $P_0$ and taking $R$ to be the third point of intersection.

Finally, note that if we are merely presented with the embedding of the elliptic curve in $\mathbb P^2$, then $P_0$ is not uniquely determined. Indeed if $3P_0 \sim H$, then we also have $3P'_0 \sim H$ for any $P'_0$ that is a $3$-torsion point with respect to the group law defined with origin $P_0$. Thanks again to Ravi for pointing this out!

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    $\begingroup$ I agree with your approach, but $P_0$ isn't determined uniquely by the condition $3P_0 \sim H$: all 3-torsion points have the same property. I think the standard thing to do is to take $P_0$ as given on an abstract elliptic curve $E$, and then embed $E$ in $\mathbb P^2$ via the linear system $|3 P_0|$. $\endgroup$ Jun 7 '18 at 1:53
  • $\begingroup$ @RaviFernando Good point - thanks! I've edited this post. $\endgroup$
    – Kenny Wong
    Jun 7 '18 at 6:58

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