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.