I am trying to piece together elliptic curves in FLT and would greatly appreciate corrections to my summary (or attempts thereof).
Mazur's paper "Number Theory as Gadfly" states, "there is a natural way of identifying lattice with a with an orbit in the complex plane" (and this would essentially be the hyperbolic uniformization?) He defines a hyperbolic uniformization to be a covering mapping from the half plane - {finite set of orbits} to an elliptic curve - {finite set of points}. Thus, he concludes that it is periodic.
We can consider an elliptic curve E to be a torus over a lattice L, because E is doubly periodic (i.e., meromorphic).
Viewing E as C/L gives information about the structure of the group of torsion points on E (according to Ribet).
Now, a torsion subgroup of E(Q) would have elements P, such that P*n=0. It is also called periodic.
So how is the torsion subgroup related to the periodicity found in elliptic curves and hyperbolic uniformizations?
Thank you!