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.
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.
- 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.
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.
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.
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?”.
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
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.