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!


Too long for a comment, eventhough this could, perhaps, be considered as kind of an answer:

First, I think it'd be clearer to write $\;nP=0\;$ and not $\;P^n=e\;$ , as the Poincare group of an elliptic curve is abelian and the operation on it is usually written additively.

Second, the "modding out by $\;n\;$" thing is not so clear to me here, as you could say the same about any group (abelian or not) with exponenent $\;n\;$ and I really don't know what this can help us out here.

Third, the "elliptic curve over a lattice" isn't clear to me, either: one can see an elliptic curve as a complex torus $\;\Bbb C/L\;,\;\;L=$ an algebraic lattice in $\;\Bbb C\;$ (i.e., a rank two free abelian subgroup of $\;\Bbb C\;$ which contains a basis for the real vector space $\;\Bbb C_{\Bbb R}\;$) , and the "equivalence" of points you talk about is just mere equality between elements represented by the very same cosets in the quotient group.

Fourth, I really have no idea what you mean by "the hyperbolic plane has periodicity, as well as a lattice..."


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