Today we discuss how some familiar objects — namely the circle, the sphere, the torus, and even elliptic curves — are each examples of non-singular algebraic curves which are also compact, connected Riemann surfaces.

### Recap

Let’s review the main definitions and concepts from last time.

Consider a prime ideal in the polynomial ring . The set is an algebraic variety. The quotient ring is an integral domain; we define as the coordinate ring of . The map which sends a point to the maximal ideal is a bijection.

We say that is an algebraic curve if . In Lecture 14, we showed that the following are equivalent:

- For each , the matrix

yields an exact sequence:

That is, the Jacobian matrix has rank while the tangent space has dimension as a complex vector space. - The Zariski cotangent space has dimension as a complex vector space for each maximal ideal .
- For each , the localization is a discrete valuation ring.
- is a Dedekind Domain.

If any of these equivalent statements holds, we say that is a nonsingular algebraic curve.

### Examples

Fix complex numbers , , , , and ; and consider the polynomial . Then the ideal in the polynomial ring is a prime ideal, so that the set is an algebraic curve over . We discuss when this is a nonsingular algebraic curve.

Define the complex numbers

Using the substitutions and we see that . Using the ideas in Lecture 5, we see that is non-singular if and only if . Such a curve is called an elliptic curve.

Now fix complex numbers , , , , and with ; and consider the polynomial in terms of the quartic polynomial . Again, the ideal in the polynomial ring is a prime ideal, so that the set is an algebraic curve over .

Define the complex numbers

Using the ideas in Lecture 5, we can find a substitution such that is equivalent to the curve . We see that is nonsingular if and only if .

### Curves as Riemann Surfaces

Now that we have considered non-singular algebraic curves, we are motivated by the following result.

Theorem.

Given a prime ideal in such that , let be a non-singular algebraic curve over . Then can be given the structure of a compact, connected Riemann surface.

We will present a general proof later, but for now we sketch the ideas with a key example. You can read a proof in Proposition 0.4 on page 4 of J. S. Milne’s course notes *Modular Functions and Modular Forms*. There is an interesting discussion on this result at math.stackexchange.com with a thread entitled “Why are Riemann surfaces algebraic curves?”.

### Riemann Sphere

Here is one example. We have seen that the Riemann sphere is a compact, connected Riemann surface. Indeed, stereographic projection is the map defined by ; it is a bijection. The open sets

form an open cover of . You may wish to compare this with Lecture 10.

### Elliptic Curves as Torii

Here is another example with a historical approach. Fix complex numbers such that . We have seen that the set is a non-singular algebraic curve. Recall that is an elliptic curve. We explain why is a Riemann surface. (We’re cheating a little here: is not compact because we’re not including the “point at infinity” . This is a small point which we will ignore for now.)

Define a function implicitly as follows: For any , define as that extended complex number such that . We call the Weierstrass pae-function after the German mathematician Karl Weierstrass (1815 — 1897). Observe that . Unfortunately this concept doesn’t make a well-defined function. Indeed, the integrand has poles at the roots , , and of the cubic , so the integral is actually path dependent. If we define the lattice in terms of the periods and , then we find that the map which sends to is a bijection. In particular,

. Using this, we say that is a torus.

In fact, using this one shows that

### Unit Circle

This function is not as strange as it might seem at first. We explain how to construct a similar map for the unit circle . Define a map implicitly by . This is not well-defined; we can only determine the values up to multiples of . In particular, is a well-defined map . The map which sends to is a familiar bijection.

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