Lecture 15: Monday, September 23, 2013

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 I = \langle f_1, f_2, \dots, f_m \rangle in the polynomial ring \mathbb C[z_1, z_2, \dots, z_n]. The set X = \left \{ P \in \mathbb A^n(\mathbb C) \, \biggl| \, f(P) = 0 \ \text{for all} \ f \in I \right \} is an algebraic variety. The quotient ring \mathcal O(X) = \mathbb C[z_1, z_2, \dots, z_n] / I is an integral domain; we define \mathcal O(X) as the coordinate ring of X. The map X \to \text{mSpec} \, \mathcal O(X) which sends a point P = (a_1, a_2, \dots, a_n) to the maximal ideal \mathfrak m_P = \langle z_1 - a_1, \, z_2 - a_2, \, \dots, \, z_n - a_n \rangle is a bijection.

We say that X is an algebraic curve if \text{dim}(X) = n - m = 1. In Lecture 14, we showed that the following are equivalent:

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

Examples

Fix complex numbers a_1, a_2, a_3, a_4, and a_6; and consider the polynomial f_1(z,w) = \bigl( w^2 + a_1 \, z \, w + a_3 \, w \bigr) - \bigl( z^3 + a_2 \, z^2 + a_4 \, z + a_6 \bigr). Then the ideal I = \langle f_1 \rangle in the polynomial ring \mathbb C[z,w] is a prime ideal, so that the set X = \left \{ (z,w) \in \mathbb A^2(\mathbb C) \, \biggl| \, f_1(z,w) = 0 \right \} is an algebraic curve over \mathbb C. We discuss when this is a nonsingular algebraic curve.

Define the complex numbers
\begin{aligned}  b_2 & = a_1^2 + 4 \, a_2 \\[5pt]  b_4 & = 2 \, a_4 + a_1 \, a_3 \\[5pt]  b_6 & = a_3^2 + 4 \, a_6 \\[5pt]  b_8 & = a_1^2 \, a_6 + 4 \, a_2 \, a_6 - a_1 \, a_3 \, a_4 + a_2 \, a_3^2 - a_4^2  \end{aligned} \qquad \begin{aligned}  g_2 & = \dfrac {b_2^2 - 24 \, b_4}{12} \\[5pt]  g_3 & = \dfrac {-b_2^3 + 36 \, b_2 \, b_4 - 216 \, b_6}{216} \\[5pt]  \Delta & = -b_2^2 \, b_8 - 8 \, b_4^3 - 27 \, b_6^2 + 9 \, b_2 \, b_4 \, b_6  \end{aligned}
Using the substitutions x = z + \dfrac {b_2}{12} and y = 2 \, w + a_1 \, z + a_3 we see that 4 \, f_1(z,w) = y^2 - \bigl( 4 \, x^3 - g_2 \, x - g_3 \bigr). Using the ideas in Lecture 5, we see that X is non-singular if and only if \Delta = g_2^3 - 27 \, g_3^2 \neq 0. Such a curve X is called an elliptic curve.

Now fix complex numbers a_0, a_1, a_2, a_3, and a_4 with a_4 \neq 0; and consider the polynomial f_1(z,w) = w^2 - f(z) in terms of the quartic polynomial f(z) = a_4 \, z^4 + a_3 \, z^3 + a_2 \, z^2 + a_1 \, z + a_0. Again, the ideal I = \langle f_1 \rangle in the polynomial ring \mathbb C[z,w] is a prime ideal, so that the set X: \, f_1(z,w) = 0 is an algebraic curve over \mathbb C.

Define the complex numbers
\begin{aligned}   g_2 & = \dfrac {4}{3} \, \bigl( a_2^2 - 3 \, a_1 \, a_3 + 12 \, a_0 \, a_4 \bigr) \\[5pt]  g_3 & = \dfrac {4}{27} \, \bigl( -2 \, a_2^3 + 9 \, a_1 \, a_2 \, a_3 - 27 \, a_0 \, a_3^2 - 27 \, a_1^2 \, a_4 + 72 \, a_0 \, a_2 \, a_4 \bigr). \end{aligned}
Using the ideas in Lecture 5, we can find a substitution such that X is equivalent to the curve y^2 = 4 \, x^3 - g_2 \, x - g_3. We see that X is nonsingular if and only if 16 \, \text{disc}(f) = g_2^3 - 27 \, g_3^2 = \Delta \neq 0.

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 I = \langle f_1, f_2, \dots, f_m \rangle in \mathbb C[z_1, z_2, \dots, z_n] such that m = n - 1, let X = \left \{ P \in \mathbb A^n(\mathbb C) \, \biggl| \, f_1(P) = f_2(P) = \cdots = f_m(P) = 0 \right \} be a non-singular algebraic curve over \mathbb C. Then X 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 X = S^2(\mathbb R) = \left \{ (u,v,w) \in \mathbb A^3(\mathbb R) \, \biggl| \, u^2 + v^2 + w^2 = 1 \right \} is a compact, connected Riemann surface. Indeed, stereographic projection is the map \sigma^{-1}: S^2(\mathbb R) \to \mathbb P^1(\mathbb C) defined by (u,v,w) \mapsto \dfrac {u + i \, v}{1 - w}; it is a bijection. The open sets
\begin{aligned}  U_1 & = \sigma \bigl( \mathbb P^1(\mathbb C) - \{ (1:0) \}  \bigr) = \left \{ (u,v,w) \in S^2(\mathbb R) \, \biggl| \, (u,v,w) \neq (0,0,+1) \right \} \\[5pt]  U_2 & = \sigma \bigl( \mathbb P^1(\mathbb C) - \{ (0:1) \}  \bigr) = \left \{ (u,v,w) \in S^2(\mathbb R) \, \biggl| \, (u,v,w) \neq (0,0,-1) \right \}  \end{aligned}
form an open cover of S^2(\mathbb R) = U_1 \cup U_2. You may wish to compare this with Lecture 10.

Elliptic Curves as Torii

Here is another example with a historical approach. Fix complex numbers g_2, \, g_3 \in \mathbb A^1(\mathbb C) such that \Delta = g_2^3 - 27 \, g_3^2 \neq 0. We have seen that the set X = \left \{ (x,y) \in \mathbb A^2(\mathbb C) \, \biggl| \, y^2 = 4 \, x^3 - g_2 \, x - g_3 \right \} is a non-singular algebraic curve. Recall that X is an elliptic curve. We explain why X is a Riemann surface. (We’re cheating a little here: X is not compact because we’re not including the “point at infinity” (0:1:0). This is a small point which we will ignore for now.)

Define a function \wp: \mathbb A^1(\mathbb C) \to \mathbb P^1(\mathbb C) implicitly as follows: For any z \in \mathbb A^1(\mathbb C), define \wp(z) \in \mathbb P^1(\mathbb C) as that extended complex number such that z = \displaystyle \int_{\infty}^{\wp(z)} \dfrac {dx}{\sqrt{4 \, x^3 - g_2 \, x - g_3}}. We call \wp the Weierstrass pae-function after the German mathematician Karl Weierstrass (1815 — 1897). Observe that \left( \dfrac {d \wp}{dz}(z) \right)^2 = 4 \, \wp(z)^3 - g_2 \, \wp(z) - g_3. Unfortunately this concept doesn’t make \wp a well-defined function. Indeed, the integrand has poles at the roots e_1, e_2, and e_3 of the cubic 4 \, x^3 - g_2 \, x - g_3, so the integral is actually path dependent. If we define the lattice \Lambda = \mathbb Z[\omega_1, \omega_2] in terms of the periods \omega_1 = \displaystyle 2 \int_{e_1}^{e_2} \dfrac {dx}{\sqrt{4 \, x^3 - g_2 \, x - g_3}} and \omega_2 = \displaystyle 2 \int_{e_1}^{e_3} \dfrac {dx}{\sqrt{4 \, x^3 - g_2 \, x - g_3}}, then we find that the map \mathbb C / \Lambda \to X which sends z to (x,y) = \bigl( \wp(z), \, \wp'(z) \bigr) is a bijection. In particular,
X \simeq \left \{ z = m \, \omega_1 + n \, \omega_2 \, \biggl| \, 0 \leq m < 1, \  0 \leq n < 1 \right \} \subseteq \mathbb C. Using this, we say that X is a torus.

In fact, using this one shows that
\begin{aligned}  g_2 & = 60 \sum_{(m,n) \in \mathbb A^2(\mathbb Z) - \{ (0,0) \}} \dfrac {1}{(m \, \omega_1 + n \, \omega_2)^4} \\[5pt]  g_3 & = 140 \sum_{(m,n) \in \mathbb A^2(\mathbb Z) - \{ (0,0) \}} \dfrac {1}{(m \, \omega_1 + n \, \omega_2)^6} \\[5pt]  \wp(z) & = \dfrac {1}{z^2} + \sum_{(m,n) \in \mathbb A^2(\mathbb Z) - \{ (0,0) \}} \left \{ \dfrac {1}{(z + m \, \omega_1 + n \, \omega_2)^2} - \dfrac {1}{(m \, \omega_1 + n \, \omega_2)^2} \right \} = \dfrac {1}{z^2} + \dfrac {g_2}{20} \, z^2 + \dfrac {g_3}{28} \, z^4 + \cdots  \end{aligned}

Unit Circle

This function \wp: \mathbb C/\Lambda \to \mathbb P^1(\mathbb C) is not as strange as it might seem at first. We explain how to construct a similar map for the unit circle S^1(\mathbb R) = \left \{ (x,y) \in \mathbb A^2(\mathbb R) \, \biggl| \, u^2 + w^2 = 1 \right \}. Define a map \wp: \mathbb A^1(\mathbb R) \to \mathbb P^1(\mathbb R) implicitly by z = \displaystyle \int_0^{\wp(z)} \dfrac {dx}{\sqrt{1 - x^2}} = \arcsin \bigl( \wp(z) \bigr). This is not well-defined; we can only determine the values up to multiples of \displaystyle 2 \int_{-1}^{+1} \dfrac {dx}{\sqrt{1-x^2}} = 2 \, \pi. In particular, \wp(z) = \sin z is a well-defined map \wp: \mathbb R / 2 \, \pi \, \mathbb Z \to \mathbb P^1(\mathbb R). The map \mathbb R / 2 \, \pi \, \mathbb Z \to S^1(\mathbb R) which sends z to (x,y) = \bigl( \wp(z), \, \wp'(z) \bigr) = \bigl( \sin z, \, \cos z \bigr) is a familiar bijection.

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About Edray Herber Goins, Ph.D.

Edray Herber Goins grew up in South Los Angeles, California. A product of the Los Angeles Unified (LAUSD) public school system, Dr. Goins attended the California Institute of Technology, where he majored in mathematics and physics, and earned his doctorate in mathematics from Stanford University. Dr. Goins is currently an Associate Professor of Mathematics at Purdue University in West Lafayette, Indiana. He works in the field of number theory, as it pertains to the intersection of representation theory and algebraic geometry.
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3 Responses to Lecture 15: Monday, September 23, 2013

  1. Pingback: MA 59800 Course Syllabus | Lectures on Dessins d'Enfants

  2. Pingback: The beauty of the Riemann sphere | cartesian product

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