Elliptic Curves, Lattices, Lie Algebras I've recently started to look at elliptic curves and have three basic questions: 


*

*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 $E$ <--> $C/L$. 

*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?

*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).
 A: For a bit more info on question 3: if you are interested in the elliptic curve only as a complex Lie group, then when you identify it with C/L for C the complex plane and L a lattice, the Lie algebra is canonically C and the exponential map is the reduction mod L.
A: Depending on what you mean, then no.  Elliptic curves (abstractly) are in bijection with lattices modulo homothety (also, the upper half plane modulo SL(2,Z)).  As for the Lie algebra of an elliptic curve, an elliptic curve is abelian, so the lie algebra is a one dimensional abelian complex Lie algebra, so it has trivial bracket.
A: *

*No, for any cubic curve in the plane, there is a family of cubic plane curves (3 or 4 dimensional - I forget Edit: 8-dimensional, with a transitive action of PGL3) that are isomorphic as curves.  For any lattice in C, there is a 1-dimensional family of lattices that form isomorphic curves.  Over C, the curves are classified by their j-invariant (a complex number).

*You can fix the lattice and choose generators so that one of the generators is 1 and the other lies in the upper half plane.  There is then an action of SL2(Z) on the choices of generators, yielding an action on the upper half plane.  If you choose a fundamental domain for this action, you get a "canonical" choice of lattice for each elliptic curve.

*The Lie algebra is the unique 1-dimensional Lie algebra, whose bracket is zero.  One can sometimes find more interesting information using the formal group law, but that mostly applies when you work in characteristic p.

A: 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).
