Elliptic Curves Without Geometry Unfortunately geometry terrifies me, so I was hoping to understand the basic theory of elliptic curves algebraically (via their function fields). Let F be a transcendence degree 1 extension of $\overline{\mathbb{Q}}$ such that F has genus 1.
(i) What are the $\mathbb{Q}$-points (e.g. in terms of the DVR's contained in F)?  
(ii) What is the meaning of the point at infinity?
(iii) What is the meaning of coordinate change between the standard charts of P$^2(\overline{\mathbb{Q}}$)?
I would be especially interested in descriptions which could be made effective.  In other words, explanations which use the word 'choose' as frequently as possible :)  But any guidance would be greatly appreciated!
 A: A $\mathbb Q$-point corresponds to a DVR whose residue field equals $\mathbb Q$.
The function field can't tell the point at infinity from any other point; the notion of "point a infinity" is not intrinsic, but is defined relative to the embedding of the curve into $\mathbb P^2$ (and a choice of coords. on $\mathbb P^2$, so that we know what the line at infinity is in $\mathbb P^2$.
The coordinate changes don't have any meaning in terms of the function field;
at best they manifest as different choices of generators.
E.g. if $E$ is given by $y^2 = x^3 + a x + b$,
then its function field $F$ equals $\mathbb Q(x)[y]/(y^2 - x^3 -ax - b ).$
If we were to change coords., say write $x= X/Z, y  = Y/Z$ in terms of homo.
coords, and then change to $u = X/Y = x/y, v = Z/Y = 1/y,$ 
so that the equation for our curve becomes $v = u^3 + a u v^2 + bv^3,$
then we would obtain the alternative expression  $F = \mathbb Q(v)[u]/
(u^3 +a uv^2 + b v^3 - v ),$ where $u = x/y,$ and $v = 1/y$.

A side remark: it's pretty hopeless to get very far in the theory of elliptic curves without thinking geometrically.  Try using your grounding in algebra as a tool and starting point for developing more geomeric intuition.  Algebraic geometry lends itself very well to this approach.  
