Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
See – rogerl May 14 '14 at 14:13
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 $Q$.) – user64687 May 14 '14 at 14:13
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 ? – mercio May 14 '14 at 14:16
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. – nigelvr May 14 '14 at 14:18
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? – nigelvr May 14 '14 at 14:19
up vote 1 down vote accepted

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.

share|cite|improve this answer
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? – nigelvr May 14 '14 at 14:21
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 $\ast$. – user64687 May 14 '14 at 14:21
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. – Brett Frankel May 14 '14 at 14:25
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. – Brett Frankel May 14 '14 at 14:33

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.