# Elliptic curves, 2-torsion and branch points.

I'm currently reading through Ravi Vakil's notes on Algebraic Geometry. I've been having trouble grasping some things conceptually though and I hope that you can help me.

For an elliptic curve (E,p) (where p is a k-valued point), we have that $\mathcal{O}_E(2p)$ has dimension 2, so we have a section that vanishes to order 2 there according to Ravi. OK, I'm fine with that, although I have to admit I'm not personally, entirely convinced. Isn't it true that all the global sections should be constant, so the other section of $\mathcal{O}_E(2p)$ (the one that is not constant) should have a pole of order 2 at p? How can it vanish then?

Now, for the second part, we have that $\mathcal{O}_E(2p)$ defines a hyperelliptic covering with 4 branch points. Why do these branch points occur from our sections? And further, If q is another branchpoint of this hyperelliptic covering, then $\mathcal{O}_E(2p) \cong \mathcal{O}_E(2q)$. I do understand that to do this, I should show that there's a section of $\mathcal{O}_E(2p)$ that vanishes at q of order 2. But how can this arise from our two sections of $\mathcal{O}_E(2p)$, one of which has divisor 2p? What goes wrong with the construction if q is not a branch point?

Many questions, but I hope that you can help me.

• Dear Dedalus, There are a lot of inaccuracies in your question: E.g. $\mathcal O_E(2p)$ is an invertible sheaf on $E$, so it doesn't have a dimension. (It has a locally free rank, which is one in this case, since it is an invertible sheaf.) Also, it is certainly not the case that all the global sections of $\mathcal O_E(2p)$ are constant. (You yourself seem to acknowledge this when you speak of the non-constant section in the subsequent clause. You are correct though that the non-constant global sections have poles of order two at $p$.) I suggest that you clarify your undestanding ... – Matt E Apr 30 '12 at 15:06
• ... of some of these foundational points before trying to understand the particular aspects of the situation under discussion. On a different note, did you try writing down an actual elliptic curve, say $y^2 = x^3 -x$, taking $p$ to be the point at infinity, and work through the discussion/construction in this case? Regards, – Matt E Apr 30 '12 at 15:07
• Dear Dedalus, Everything in your comment is correct. In particular, the section $1$ of $\mathcal O_E$, when regarded as a section of $\mathcal O_E(2p)$, vanishes to order $2$ at $p$ (just as you wrote). But this is not the section that is going to give a finite map to $\mathbb P^1$; for that you need a non-constant rational function, and so a non-constant section of $\mathcal O_E(2p)$. Regards, – Matt E Apr 30 '12 at 15:41
• Dear Dedalus, Did you try looking at a concrete example? Regards, – Matt E May 1 '12 at 0:30
• Dear Matt, I believe I tried to check with an example.But probably I'm misunderstanding something. Let us take y^2=x^3-x, so that p would be at infinity. Then, we have that the fraction field will be k(x)[y]/(y^2-x^3-x). So, if we look at say (x-1) and wanna find the divisors of zeros and poles, we first see that if we localize at $(x-1,y)$, y will be a local uniformizer, and since (x-1) = y^2/(x+1)x, it will have a zero of order 2 there (?) and for infinity, we do a similar analysis, and see that it has a pole of order 2? Am I at all correct or totally wrong, or somewhere in between? – Dedalus May 1 '12 at 10:25