On the group structure of elliptic curves

May 11, 2014

In last post we proved Abel’s theorem and as a corollary we saw that any compact Riemann surface $S$ of genus $g=1$ (a smooth elliptic curve) is biholomorphic to a complex torus, the biholomorphism being given by the Abel-Jacobi mapping:

$\mu|_S : S \overset{\cong}\longrightarrow \mathbb{C}/\Lambda$

which is given by

$\mu(p) = \displaystyle\int_{p_0}^p\omega$ (mod $\Lambda$ )

where some $p_0 \in S$ , $\omega \in \Omega^1(S)$ is a holomorphic 1-form and $\Lambda$ is the period lattice $\Lambda = \{\sum_{i=1}^2m_i\Pi_i \mid m_i \in \mathbb{Z}\}\subset \mathbb{C}$ for $\Pi_i$ the two period vectors $\Pi_i = \int_{\delta_i}\omega \in \mathbb{C}$ for $\delta_1, \delta_2$ giving a basis for $H_1(S,\mathbb{Z})$ .

In particular, every such Riemann surface $S$ has a group structure and we will see that this description of the group structure on an elliptic curve is the same as the usual one given for elliptic curves in $\mathbb{P}^2$ given by a cubic polynomial. By the usual group structure I mean the following: recall that by Bézout’s theorem, if you intersect a line in $\mathbb{P}^2$ with the zero locus of a cubic polynomial, you get (counting multiplicities) exactly 3 points. Thus to add 2 points $p_1, p_2$ , you consider the line between them and declare that $p_1+p_2 + p_3 = 0$ if $p_3$ is the third point of intersection of this line and the elliptic curve.

After a preliminary discussion of projective embeddings of Riemann surfaces, we will see that in fact any genus 1 compact Riemann surface can be embedded in $\mathbb{P}^2$ as the zero locus of some cubic polynomial. In particular when the initial surface is a complex torus $\mathbb{C}/\Lambda$ , this embedding is given by the so-called Weierstrass $\wp$ -functions. We will then see that this embedding is actually an inverse to the Abel-Jacobi mapping.

First the remarks on projective embeddings of compact Riemann surfaces. Suppose $L \to M$ is a holomorphic line bundle over a compact complex manifold $M$ and $s_0, \ldots, s_N$ are a basis for $V = H^0(S,\mathcal{O}(L))$ the global holomorphic sections of $L$ . Suppose that there is no point on $M$ such that every sections in $V$ vanish simultaneously. Then for every $p \in M$ , we have a non-zero vector

$(s_0(p), \ldots, s_N(p)) \in (L_p)^{N+1}$ .

By choosing a trivialisation, we can consider this vector as sitting in $\mathbb{C}^{N+1}$ . Of course this vector will depend on the choice of trivialisation of $L$ around $p$ but different choices of trivialisations will only change the vector by a complex multiple. We thus get a map

$\iota_L : M \to \mathbb{P}^N$ defined by $\iota_L(p) = [s_0(p):\ldots:s_N(p)]$ .

Kodaira’s embedding theorem states that when $L \to M$ is a positive line bundle (i.e. admits a connection of positive curvature), there exists a $k_0 \in \mathbb{N}$ such that for $k \geq k_0$ , the map

$\iota_{L^k} : S \to \mathbb{P}^N$

is well-defined (the global sections of $\mathcal{O}(L)$ are never all vanishing) and is an embedding of $M$ .

We can in fact give a sharper version of Kodaira’s embedding theorem for compact Riemann surfaces (you can find a detailed discussion of this on pages 214, 215 of Griffiths-Harris):

Theorem: Let $S$ be a compact Riemann surface and $L \to S$ a holomorphic line bundle. If $\deg L > \deg K_S + 2$ , then $\iota_L : S \to \mathbb{P}^N$ is well-defined and an embedding.

In this proposition, the degree of a line bundle is just $c_1(L)$ under the identification $H^2(S,\mathbb{Z}) \cong \mathbb{Z}$ or equivalently, since if $[\sum_ia_ip_i] = L$ then the Poincaré dual $\eta_D = c_1(L)$ , the degree is $\sum_ia_i$ . Also, $K_S \cong \bigwedge^{1,0}T^*S$ is the canonical bundle of $S$ and $N = \dim H^0(S,\mathcal{O}(L)) - 1$ .

The case that will interest us is when the genus of $S$ is $g=1$ . What we show is that in fact we can take $N=2$ . Indeed, since $K_S = 0$ because $T^*S$ is trivial ( $S$ is topologically a torus), for any $p \in S$ the above theorem tells us that for the bundle $L = [3p]$ , the map $\iota_L$ gives an embedding of $S$ into $\mathbb{P}^N$ for $N = h^0(S, \mathcal{O}(L)) - 1.$ If $s \in H^0(S,\mathcal{O}([p]))$ , then $s,s^2,s^3 \in H^0(S,\mathcal{O}(L))$ are 3 linearly independent global holomorphic sections so $N \geq 2$ . On the other hand, a global section of $\mathcal{O}(L)$ corresponds to a meromorphic function on $S$ having poles only at $p$ and at worst of degree $3$ . Such a function is completely determined by the first four coefficients in its Laurent expension around $p$ :

$\dfrac{a_{-3}}{z^3} + \dfrac{a_{-2}}{z^2} + \dfrac{a_{-1}}{z} + a_0 + \cdots$

(the difference of two such functions having all of these 4 numbers equal would be a global holomorphic function vanishing at $p$ hence everywhere). Moreover, two such functions having the same $a_{-3}$ and $a_{-2}$ must have the same $a_{-1}$ because otherwise their difference would be a meromorphic function having only one pole at $p$ and of order 1, which is impossible if the genus is not zero. So we need at most 3 parameters to determine such a function, hence $h^0(S,\mathcal{O}(L) \leq 3$ . This shows $N=2$ .

We can write this embedding explicitly as follows: by Kodaira’s vanishing theorem, we have $H^1(S,\mathcal{O}(p)) = 0$ so considering the long exact sequence associated to the short exact sequence

$0 \to \mathcal{O}(p) \to \mathcal{O}(2p) \overset{res}\to \mathbb{C}_p \to 0$

where $\mathbb{C}_p$ is the skyscraper sheaf at $p$ , we find $H^0(S,\mathcal{O}(2p)) \to \mathbb{C} \to 0$ so there exists a meromorphic function $F$ on $S$ having a double pole at $p$ and no other poles. Since $h^0(S,\Omega^1) = g = 1$ , there is also on $S$ a nonzero holomorphic 1-form $\omega$ . But as remarked earlier, since the canonical bundle $K_S$ is trivial, the divisor $(\omega)$ has zero Poincaré dual so is itself zero, i.e. $\omega$ is nowhere vanishing. We consider $F\omega$ . It is a meromorphic form with only one pole at $p$ , of order 2, and since the sum of the residues of a meromorphic 1-form must be zero, $\text{res}_p(F\omega) = 0$ .

So if $z$ is a local holomorphic coordinate near $p$ , we can write

$F(z) = \dfrac{1}{z^2} + a_1z + a_2z^2 + \cdots$ near $p$

after possibly multiplying by a constant and adding a constant. Considering also the meromorphic function $dF/\omega$ , which is holomorphic except at $p$ where it has a triple pole, we get a global meromorphic function $F' = \lambda(dF/\omega) + \lambda'F + \lambda''$ which is

$F'(z) = \dfrac{1}{z^3} + a_1'z + a_2'z^2 + \cdots$ near $p$

for the right choice of $\lambda$ ‘s. Since $1, F$ and $F'$ are three linearly independant global sections of $\mathcal{O}([3p])$ , they form a basis and thus the embedding $\iota_L : S \to \mathbb{P}^2$ is given by

$\iota_L(q) = [1:F(q):F'(q)]$ .

We can use this to explicitly describe $S$ as the zero set of a polynomial in $\mathbb{P}^2$ : around $p$ , we have

$F'(z)^2 = \dfrac{1}{z^6} + \dfrac{c}{z^2} + \dfrac{a}{z} + \cdots$

and

$F(z)^3 = \dfrac{1}{z^6} + \dfrac{c'}{z^3} + \dfrac{c''}{z^2} + \dfrac{a'}{z} + \cdots$ .

Thus the meromorphic function $F'(z)^2 + c'F'(z) - F(z)^3 + (c''-c)F(z)$ has at most a simple pole at $p$ and no other poles, so is constant. This tells us that $\iota_L(S)$ is included in the zero locus of the polynomial $y^2 + c'y = x^3 + ax + b$ which can be rewritten by making suitable linear changes in the coordinates by

$y^2 = x(x-1)(x-\lambda)$

for $\lambda \in \mathbb{C}$ . Note that here $(x,y)$ is a set of affine coordinates in $\mathbb{P}^2$ on the open set $\{[z_0,z_1,z_2] \mid z_0 \neq 0\}$ . Since there are both topological torus (the second by the degree-genus formula), they must be equal, i.e. the zero locus of this polynomial in $\mathbb{P}^2$ is exactly $\iota_L(S)$ .

We will show that in fact the above construction gives an inverse for the Abel-Jacobi map $\mu : \mathbb{P}^2 \supset S \overset{\cong}\to \mathbb{C}/\Lambda$ which realises such a non-singular planar curve as a complex torus.

Consider $S = \mathbb{C}/\Lambda$ and $F,F'$ as above, with their pole at $p_0 \in S$ . Take $\omega = dF/F'$ as a basis for $H^0(S,\Omega^1)$ and $z$ the coordinate on $S$ such that $\omega = dz$ . In this context, the function $F$ is the so-called Weierstrass $\wp$ -function. Its derivative $(\partial / \partial z)\wp = -2F'$ is denoted $\wp'$ . If the Laurent expansions of $\wp$ around $p_0$ contained a term of odd degree, then killing the order 2 pole by $\wp(-z) - \wp(-z)$ we would obtain a noncontant holomorphic function on $S$ . So it has no term of odd degree and we can write

$\wp(z) = \dfrac{1}{z^2} + az^2 + bz^4 + \cdots$

$\wp'(z) = -\dfrac{2}{z^3} + 2az + 4bz^3 + \cdots$

$\wp(z)^3 = \dfrac{1}{z^6} + \dfrac{3a}{z^2} + 3b + cz^2 + \cdots$

$\wp'(z)^2 = \dfrac{4}{z^6} - \dfrac{8a}{z^2} - 16b + c'z + \cdots$ .

We find the relation

$\wp'^2 = 4\wp^3 - g_2\wp - g^3$ ,

where $g_2 = 20a$ and $g_3 = 28b$ . The corresponding embedding is then

$\psi : \mathbb{C}/\Lambda \to \mathbb{P}^2$ where $\psi(z) = [1:\wp(z):\wp'(z)]$ ,

and $\psi(\mathbb{C}/\Lambda)$ is the zero locus of the polynomial

$y^2 = 4x^3 - g_2x - g_3$ ,

written in suitable affine coordinates. On the other hand, the Abel-Jacobi map from $\psi(S)$ to $\mathbb{C}/\Lambda$ is given by

$\mu(p) = \displaystyle\int_{p_0}^p\frac{dx}{y}$ (mod $\Lambda$ )

with let’s say $p_0 = \psi(0)$ . Indeed, $\psi^*(dx/y) = \wp'(z)dz/\wp'(z) = dz = \omega$ so $dx/y$ is a non-zero holomorphic 1-form on $S$ . Moreover, $\mu|_S$ is actually inverse to the embedding $\psi$ because

$\mu(p) = \displaystyle\int_{\psi(0)}^p \frac{dx}{y} = \int_0^{\psi^{-1}(p)}dz = \psi^{-1}(p)$ .

We have thus found an inverse for the Abel-Jacobi mapping of a smooth elliptic curve in $\mathbb{P}^2$ .

We are now in a position to discuss the group structure from different point of views. Any Riemann surface $S$ of genus $g=1$ inherits a group structure simply by letting

$p_1+_{\mu}p_2 = \mu^{-1}(\mu(p_1) + \mu(p_2))$ for $p_1, p_2 \in S.$

There is also a group structure induced by an embedding $\psi : S \hookrightarrow \mathbb{P}^2$ as above : $p_1 +_{\psi} p_2 +_{\psi} p_3 = 0$ if $\psi(p_1),\psi(p_2),\psi(p_3)$ are colinear in $\mathbb{P}^2$ . These are in fact the same. To see this, consider $S$ as embedded in $\mathbb{P}^2$ , take 3 points $p_1,p_2,p_3 \in S$ and denote $z_i = \mu(p_i) \in \mathbb{C}/\Lambda$ the corresponding points in the complex torus. Then by Abel’s theorem,

$z_1 + z_2 + z_3 = 0$ iff $\exists f \in \mathcal{M}(S)$ with $(f) = p_1 + p_2 + p_3 - 3p_0 = 0$ .

But suppose $p_1 +_{\psi} p_2 +_{\psi} p' = 0$ for $p' \in S$ , i.e. with $A(x,y) = ax + by + c$ the line joining $p_1, p_2$ (in affine coordinates), we also have $A(p') = 0$ . Then

$g(z) = A(\wp(z),\wp'(z)) = a\wp(z) + b\wp'(z) + c \in \mathcal{M}(\mathbb{C}/\Lambda)$

is a meromorphic function on the torus having divisor $(g) = z_1 + z_2 + \mu(z') - 3.0$ so the function $f = g\circ \mu \in \mathcal{M}(S)$ has divisor $(f) = p_1 + p_2 + p' - 3p_0$ . Thus if $p_1, p_2, p_3$ are collinear, we have $p_1 +_{\mu} p_2 +_{\mu} p_3 = 0$ and conversely, if there is a meromorphic function $h$ on $S$ such that $(h) = p_1 + p_2 + p_3 - 3p_0$ , then $h-g$ is a meromorphic function with $(h-g) = p_3 - p'$ and this forces $p_3 = p'$ because otherwise we would have a meromorphic function having a simple pole at $p'$ and no other poles.