This exposition introduces how one can naturally come across elliptic curves by exploring rather natural questions in complex analysis.

Doubly periodic functions

One recalls that in real analysis, given a square intergrable function with period , i.e. , one can write express as a sum of , in particular,

with some appropriate notion of convergence. One could ask a similar question over the complex numbers, but in this case since is 'two dimensional', we can ask for double periodicity, i.e. . If the function is simply integrable, one can perform a similar fourier transform so it doesn't get much more interesting. However what if we require that the function is complex differentiable, or perhaps meromorphic? The answer turns out to be rather elegant and leads us naturally to considering elliptic curves!

Let be the lattice generated by and be the fundamental domain. One viewpoint we can take is that is a function from where the map is a map of Riemann surfaces. We will go back to this viewpoint eventually.

First, let's search for nontrivial examples of such functions. The simplest example one can come up with is $$\sum_{\lambda\in\Lambda}(z-\lambda)^{-3}$$ where this sum converges absolutely so the order of the summation doesn't matter. (note as long as the exponent is smaller than it works). It turns out that this is not quite the correct function to consider, instead we integrate this function and throw in some constant terms to obtain the weiestrass elliptic functions:

Similarly one can verify both sum converges absolutely, in the first case since the term in the sum grows like eventually, hence order of summation is not a concern here either. To verify double periodicity, notice that is doubly periodic and , hence it follows that is doubly periodic.

It turns out that if is doubly periodic, it can be written as a rational fucntion of and ! More specifically, if is even and doubly periodic, we can write it as a rational function in terms of . The general case follows using the fact is odd.

First, we show that has finitely many zeros/poles in . Suppose it has infinitely many, then since is compact, the set of all zeros/poles cannot be isolated but that contradicts the memomorpic assumption.

Since we know has finitely many zeros/poles, we can construct some rational function such that has the same zeros/poles as counting multiplicity since we can't have essential singularities (again the meromorphic condition). Since has no poles or zeros, it is bounded on , hence on the entire complex plane, but this implies it is constant!

Elliptic curves

Since the proof above is constructive, and we can apply it to the even function to obtain

Now by considering the laurent series expansion and defining

for , one obtains

which looks precisely like the elliptic curves we see often in cryptography or other areas that you may encounter them in! We typically define so our curve now looks like

With this, we have an extremely simple way to add points on the curve :

It turns out that the solutions to the elliptic curve corresponds bijectively to , more specifically, the map

is a bijection from to solutions to . The bijectivity comes from noticing that is ramified double covering with four branch points since from and the branch points are .

Let , then one can prove that if , then are colinear. Notice that for any doubly periodic meromorphic function , both the sum of orders of poles/zeros as well as sum of the positions of the poles/zeros must add up to zero in by integrating and respectively. Let be on the line , then since has a triple pole at and a zero at and respetively, the last zero bust be at , hence we are done. The case of or at infinity is handled similarly. This leads to a natural way to generalize addition of points on elliptic curves to other fields!

By considering doubly periodic meromorphic functions on the complex plane, one obtains elliptic curve and the point addition formula with a sufficient number of application of Cauchy's residue theorem. It turns out one can do a lot more with this! By studying maps between different lattices, one can obtain isogenies and complex multiplication quite easily. By trying to compute for , one quickly finds modular forms appearing. The construction that points on a line sum to zero is precisely the picard group, which leads to a wild list of constructions in algebraic geometry!

References

• Neal Koblitz - Introduction to Elliptic Curves and Modular Forms
• Fred Diamond and Jerry Shurman - A First Course in Modular Forms