I've recently started to look at elliptic curves and have three basic questions: 


1. Is it correct to say that elliptic curves $E$ in the projective plane are in **bijective** correspondence with lattices $L$ in the complex plane via $E$ <--> $C/L$.
2. If so, is there an explicit expression of the lattice generators in terms of the equation defining the curve? Or, at least, is there a simple example of a curve and its corresponding lattice?
3. Since every elliptic curve is a Lie group, it must have a corresponding Lie algebra. Is there an explicit expression of the Lie algebra in terms of the equation or lattice? Or, again, a simple example of a curve and its Lie algebra (or, even better, an example of a curve, its lattice, and its Lie algebra).



Elliptic Curves, Lattices, Lie Algebras
What you want in terms of the relation between lattices and elliptic curves over C is proposition I.4.4 of Silverman's Advanced topics in the arithmetic of elliptic curves. Additionally, to go from a lattice to the equation of the elliptic curve (explicitly), you use Eisenstein series as in Corollary I.4.3 of that book. To go from the elliptic curve to the lattice is described in the proof of proposition I.4.4: you look at the homology of the curve and compute period integrals (incidentally, this is how you go from an abelian variety over C to a complex torus).


