## 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:

• For each $P \in X$, the $m \times n$ matrix
$\text{Jac}_P(X) = \left[ \begin{matrix} \dfrac {\partial f_1}{\partial z_1}(P) & \dfrac {\partial f_1}{\partial z_2}(P) & \cdots & \dfrac {\partial f_1}{\partial z_n}(P) \\[10pt] \dfrac {\partial f_2}{\partial z_1}(P) & \dfrac {\partial f_2}{\partial z_2}(P) & \cdots & \dfrac {\partial f_2}{\partial z_n}(P) \\[10pt] \vdots & \vdots & \ddots & \vdots \\[5pt] \dfrac {\partial f_m}{\partial z_1}(P) & \dfrac {\partial f_m}{\partial z_2}(P) & \cdots & \dfrac {\partial f_m}{\partial z_n}(P) \end{matrix} \right]$
yields an exact sequence:
$\{ 0 \} \longrightarrow T_P(X) \longrightarrow \mathbb A^n(\mathbb C)\overset{\text{Jac}_P(X)}{\longrightarrow} \mathbb A^m(\mathbb C) \longrightarrow \{ 0 \}.$
That is, the Jacobian matrix $\text{Jac}_P(X)$ has rank $m$ while the tangent space has dimension $\text{dim}_{\mathbb C} \bigl( T_P(X) \bigr) = \text{dim}(X)$ as a complex vector space.
• The Zariski cotangent space has dimension $\text{dim}_{\mathbb C} \bigl( \mathfrak m / {\mathfrak m}^2 \bigr) = \text{dim}(X)$ as a complex vector space for each maximal ideal $\mathfrak m \in \text{mSpec} \, \mathcal O(X)$.
• For each $P \in X$, the localization $\mathcal O_P$ is a discrete valuation ring.
• $\mathcal O(X)$ is a Dedekind Domain.

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.