Let $C$ be an elliptic curve over a field $k \supset \mathbb{Q}$. Then given $P$ and $Q$, we can draw the line between $P$ and $Q$ (call this line $L$) and then "find the third intersection point", and call this point $P \ast Q$.

If neither $P$ nor $Q$ lie on the tangent line to $C$, then each $P$ and $Q$ have intersection multiplicity $1$, so by Bezout's theorem, we can find a third point, $Q$, not equal to $P$ or $Q$.

What do we take to be our third point if either $P$ or $Q$ lie on the tangent to $C$? (e.g. if $L$ is the tangent to $C$ at $P$, then $P$ has intersection multiplicity at least two and we don't get a third distinct point).

edit: $\ast$ here is not the group law, it's just the third intersection point operation.

  • $\begingroup$ See en.wikipedia.org/wiki/Elliptic_curve $\endgroup$ – rogerl May 14 '14 at 14:13
  • $\begingroup$ If $L$ is tangent to $C$ at $P$, and $Q$ is different from $P$, then the third point on $L \cap C$ is $P$. (Likewise if $L$ is tangent at $Q$.) $\endgroup$ – user64687 May 14 '14 at 14:13
  • $\begingroup$ what do you mean with "the tangent line to $C$" ? there are lots of tangent lines to $C$. And what does "$L$ goes through the tangent" means ? $\endgroup$ – mercio May 14 '14 at 14:16
  • $\begingroup$ mercio: at a point $P$, there is a unique tangent line to $P$ on $C$, as $C$ is smooth. The other thing was a typo. $\endgroup$ – nigel May 14 '14 at 14:18
  • $\begingroup$ Asal Beag Dubh: OK, I suppose that's just a definition? If $L$ is tangent to $C$ at $P$, why not take $Q$ to be the third intersection point? $\endgroup$ – nigel May 14 '14 at 14:19

If the line is tangent to $C$ at $P$ and also passes through $Q$, then think of the line as passing through $P$ twice and through $Q$ once, so $P*Q=P$. That is, $P$ is the third point on the line. Similarly, $P*P=Q$ in this situation.

If the line only intersects $P$, then $P$ is a triple point and $P*P=P$.

It is probably worth noting that there is a group law on $C$ which is similar to but not quite as you have described it above. That is to say, there is a better addition rule on $C$ as follows: If $L\cap C=\{P,Q,R\}$, we define $P + Q=-R$. For reasons I won't get into here, this addition rule has much better properties.

  • $\begingroup$ OK, thanks. But if $L$ is tangent to $C$ at $P$, then I agree that $L$ passes through $P$ twice and $Q$ once ... but then why not take $Q$ to be the third point of intersection? Is this just part of the definition? $\endgroup$ – nigel May 14 '14 at 14:21
  • $\begingroup$ Dear Brett, I think your correction in the final paragraph is unnecessary: the OP did not say that $\ast$ was actually the group operation. Indeed, it is very common in sources I am familiar with to denote this "third point on the line" operation by $\ast$. $\endgroup$ – user64687 May 14 '14 at 14:21
  • $\begingroup$ Dear @AsalBeagDubh I agree that it is unnecessary, but I hope that it is still informative. I'll adjust my notation to try to avoid any confusion. $\endgroup$ – Brett Frankel May 14 '14 at 14:25
  • $\begingroup$ Dear @nigelvr It depends which points you are combining. If you want $P*Q$, then $P$ is the remaining point. If you want $P*P$, then $Q$ is the remaining point. You should count each point once for each time it appears on either side of the $*$; this way there will be exactly one point left over. $\endgroup$ – Brett Frankel May 14 '14 at 14:33

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