Elliptic Curve Theory

An abstract elliptic curve is defined to be a "smooth projective algebraic curve of genus over a field ". No idea what that means (yet). In this note I focus on elliptic curves over the complex field .

As the word "projective" implies, these curves intersect the line at infinity at a special distinguished point . To find out what this point is, we can consider an elliptic curve where has characteristic not 2 or 3 (in the case the characteristic is 2 or 3, I cannot figure out how the point is calculated). This allows us to represent in Weierstrass form:

which on converting to homogeneous coordinates gives:

To find the intersection of this with the line at infinity, we set , and then by properties of fields we find that triples of the form satisfy the resulting equation, which represents the point in projective space. This is our distinguished point .

It is important to note that elliptic curves over finite fields do have a lot of practical use. Most notably, they are used in elliptic curve cryptography.

Complex Elliptic Curves

Any complex elliptic curve can be expressed in Weierstrass Form:

since the characteristic of the base field ( ) is 0. These elliptic curves have a close relation to lattices in the complex plane and complex tori. This in turn gives us a deep relationship between complex elliptic curves and the modular group .

Complex Tori

Given numbers that are -linear independent, ie. for any , we can form a lattice

This lattice forms an abelian group (under addition), basically by definition. Since forms an abelian group under addition, automatically becomes a normal subgroup, hence we can now form the quotient group . This quotient has both the structure of an abelian group and a topological complex torus, as shown below.

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As a Riemann surface, we can define meromorphic functions on the space , and one such important function is the Weierstrass function. The details of the definition of this function are beyond me right now.

function and elliptic curves

Some complex analysis magic implies that the function satisfies the differential equation:

which is really close to the form of a complex elliptic curve. However, for us close enough is good enough; given a complex elliptic curve

we can perform the substitution to obtain a curve

Which exactly matches the form of the differential equation, giving us and . Hence we obtain a mapping from lattices in to elliptic curves via the mapping (recall that the ambient space is a projective space).

If you thought this was slightly cheating because we used a parametrization to force the original Weierstrass form into the form of the differential equation, well consider the fact that we are viewing these through the lens of algebraic geometry. In the relevant category for these curves, which is the category of pointed smooth projective curves over and regular maps of pointed curves, our substitution is actually an isomorphism of projective curves. Hence for all intents and purposes, these two forms are "the same", and we really can think of the Weierstrass function giving a correspondence between lattices in and complex elliptic curves.

Moduli space of complex elliptic curves

Given an elliptic curve , we have an associated lattice . Fixing a complex number , the mapping is a biholomorphic map that descends to a biholomorphic isomorphism , which has just the right properties to induce an isomorphism of elliptic curves.

Putting on our category theory glasses, we only care about objects up to isomorphism, so we can identify with , and then we see that

Hence every lattice is determined uniquely (up to isomorphism) by a single complex number. Additionally we can make the assumption that , the complex upper half plane because if it were not, we could just divide by at the start to fix it.

So now what we have is a lattice determined by a single complex number that corresponds to our original elliptic curve via the previous Weierstrass function business. But this mapping is still not injective (even if considered to isomorphism classes of elliptic curves).

The modular group has an action on the complex upper half plane via Möbius transformations. Denoting this action by for , we can now note the surprising deep connection between our previous theory and this action:

A lattice and the lattice for correspond to the same (complex) elliptic curve (up to isomorphism). This opens up an isomorphism between two (seemingly) unrelated structures:

that the moduli space of complex elliptic curves ( ) is isomorphic to the quotient of the upper half plane by the action of the modular group ( ).