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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.

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