Elliptic functions, part II

This post is a continuation of my previous post about elliptic functions.

We showed that the {\wp} function which is {\Omega}-invariant satisfies the differential equation

\displaystyle f'(z)^2=4f(z)^3-g_2f(z)-g_3.

where the constants g_n are given in terms of

\displaystyle \sum_{\omega \in \Omega^*}\frac{1}{\omega^{2n}}

We did this by neutralizing the only pole of {\wp'(z)^2} on {E=\mathbb{C}/\Omega}, by adding to {\wp'(z)^2} a suitable polynomial in {\wp(z)}.

Thus we can use the functions {\wp(z), \wp'(z)} to parametrize the curve

\displaystyle y^2=4x_3-g_2x-g_3

in {\mathbb{C}^2}. In fact we’re really parametrizing the projective curve

\displaystyle \tilde E = V(Y^2Z-4X^3-g_2XZ^2-g_3Z^3)

in {\mathbb{P}^2(\mathbb{C})} by using the map

\displaystyle \Psi : E \rightarrow \tilde E

\displaystyle z \mod \Omega \mapsto \begin{cases} (\wp(z), \wp'(z), 1) & \text{if }(z \mod \Omega) \neq 0 \\ (0,0,1) & \text{otherwise.}\end{cases}

What we’re doing is exactly analogous to the parametrization of a conic using trigonometric functions.

With a bit more work, we can see that the field of elliptic functions with respect to {\Omega} is precisely the abstract field {\mathbb{C}(x,y)}, subject to the relation {y^2=4x^3-g_2x-g_3} (i.e. the quotient field of {\mathbb{C}[x,y]/(y^2-4x^3-g_2x-g_3)}). This means that for each {\Omega}-elliptic function {f}, we can construct a rational function of {\wp} and {\wp'} which has the same poles and zeroes, and thus express {f} as a rational function in {\wp} and {\wp'}.

Now since {\mathbb{C}} is algebraically closed, we can factor our equation as

\displaystyle \wp'(z)^2=\wp(z)^3-g_2\wp(z)-g_3=(\wp(z)-e_1)(\wp(z)-e_2)(\wp(z)-e_3)

for suitable values of {e_i}. It’s easy to see from this equation that {e_i=\wp(c_i)}, where {c_i} runs over the zeroes of {\wp'(z)}. Counted with multiplicities, there are {3} points where {\wp'(z)} vanishes, since {\wp(z)} of degree {3} as a cover of {\mathbb{P}^1(\mathbb{C})}. Using the fact that {\wp'(z)} is an odd function and periodic with respect to {\Gamma}, we can see that {\wp'(z)} vanishes at the symmetry points of the fundamental paralellogram having coordinates

\displaystyle \frac{\omega_1}{2}, \frac{\omega_2}{2}, \frac{\omega_1+\omega_2}{2}.

Thus we have

\displaystyle (e_1, e_2, e_3)=\left(\wp\left(\frac{\omega_1}{2}\right), \wp\left(\frac{\omega_2}{2}\right), \wp\left(\frac{\omega_1+\omega_2}{2}\right)\right).

Moreover, these three values are distinct. Indeed, it’s easy to see that each one is taken with multiplicity two by construction (i.e. each is a double zero of {\wp'(z)}), and since {\wp} takes each value exactly twice, no two of them can be equal. To see that these points are double points, notice that the derivative of {\wp(z)-\wp(\omega_1/2)} vanishes at {\omega_1/2}, so the point {\frac{\omega_1}{2}} is a double point. This implies that the discriminant of {f(x)=(x-e_1)(x-e_2)(x-e_3)} does not vanish, which implies after a quick check that the curve {\tilde E}, which is the locus of zeroes of {Y^2Z-4X^3-g_2XZ^2-g_3Z^3} in the projective plane, is actually a nonsingular curve. (From now on we’ll call both {\tilde E}, {E} curves.)

So we have two curves: {E} is defined in an analytic way, because its function field is constructed as a subfield of the field of meromorphic functions on {\mathbb{C}}. On the other hand, the curve {E} is an algebraic curve.

In fact, the curves {E} and {\tilde E} are exactly the same in every respect, as it turns out. This means that the curve {\tilde E} can be made into a group, since the curve {\mathbb{C}/\Omega} is naturally a group (it’s just the torus group). Of couse, the magical thing that happens is that the group law on {\tilde E} has a beautiful geometric interpretation, and that it’s given by rational functions on {\tilde E}.

Let’s compare again with trigonometric functions. Consider the locus {S} of {x^2+y^2=1} in {\mathbb{C}^2}. We know how to add points on the (usual) circle by adding angles. We can prove by elementary geometry, or using the series definitions of trigonometric functions, the formula

\displaystyle (x,y)+(x',y')=(xx'-yy', xy'+x'y),

which shows that the group structure on {S} is given by rational (polynomial!) functions. What is amazing is that the group structure is compatible with the {(\sin t, \cos t)} parametrization of {S} – in fact, the group law becomes a pair of “addition theorems”: one for {\sin} and one for {\cos}.

For the rest of this post, I will assume that the reader is familiar with the simple geometric interpretation of the group law on an elliptic curve. For an easy description, see the wikipedia page.

So it is easy to “discover” the addition theorem for {\wp} if we take the group law on {\tilde E} for granted. By a simple calculation, we obtain that, for {z \neq Z},

\displaystyle \wp(z+Z)+\wp(z)+\wp(Z) = \left(\frac{\wp'(z)-\wp'(Z)}{\wp(z)-\wp(Z)}\right)^2.

For example, to get the formula for {\wp'(z+Z)}, we find the {y}-coordinate of the point

\displaystyle (\wp(z), \wp'(z))+(\wp(Z), \wp'(Z))

using the geometric law. To ease notation a bit, let

\displaystyle P=(x_1, y_1)=(\wp(z), \wp'(z))

\displaystyle Q=(x_2, y_2) = (\wp(Z), \wp'(Z))

\displaystyle P*Q=(x_3, y_3)

Now the line passing through {P} and {Q} (assuming they are distinct, so {z} and {Z} are distinct points) is {y-y_1=\lambda(x-x_1)}, where {\lambda = (y_1-y_2)/(x_1-x_2)}. We substitute this value of {x=\lambda^{-1}(y-y_1)+x_1} in the equation {y^2=4x^3-g_3x-g_2} and we get the cubic in {y}

\displaystyle \lambda^3y^2=4(y-y_1-\lambda x_1)^3-g_2\lambda^2(y-y_1-\lambda x_1) - g_3\lambda^3.

Which is, after dividing by {4} and rearranging terms,

\displaystyle y^3-y^2(\lambda^3-3(y_1+\lambda x_1))-\dots =0

Now we already know two roots of this cubic; they are the {y}-coordinates of {P} and {Q}, by construction. Thus, by inspecting the coefficient of {y^2} in this cubic, which is {-(y_1+y_2+y_3)}, we see that

\displaystyle (y_1+y_2+y_3)=3(y_1+\lambda x_1)-\lambda^3

and hence, by the definition of addition on the elliptic curve {E:\ y^2=4x^3-g_2x-g_3} and by the (still unjustified) assumption that the group structure is compatible with the coordinates {(\wp, \wp')}, that the function {\wp'} satisfies the addition theorem

\displaystyle -\wp'(z+Z)=3\left(\wp'(z)+\wp(z)\frac{\wp'(z)-\wp'(Z)}{\wp(z)-\wp(Z)}\right)-\left(\frac{\wp'(z)-\wp'(Z)}{\wp(z)-\wp(Z)}\right)^3

\displaystyle =3\frac{\wp'(z)\wp(Z)-\wp'(Z)\wp(z)}{\wp(z)-\wp(Z)}-\left(\frac{\wp'(z)-\wp'(Z)}{\wp(z)-\wp(Z)}\right)^3

As expected, this expression is symmetric in {z} and {Z}. The doubling formula, i.e. the case {z=Z}, is obtained by taking the limit as {z \rightarrow Z} in the addition theorem.

Of course, none of this is justified because we haven’t explained why the coordinates {(\wp, \wp')} should be compatible with the group structure. In fact, it makes much more sense to think of the group structure on the elliptic curve as a consequence of the addition theorems. So, to understand why the group structure really is what it is, we have to understand where these addition theorems really come from.

Recall from complex analysis that, for a function {f} meromorphic on a domain {D}, the integral

\displaystyle \int_{\delta D} \frac{df}{f}

equals {2\pi i (Z-P)}, where {Z} and {P} denote the number of zeroes and poles of {f} on {D}, each taken with appropriate multiplicity.

By multiplying the differential {df/f} by a function {g} holomorphic on {D}, we obtain a weighted sum over the zeroes and poles of {f}. More precisely,

\displaystyle \frac{1}{2\pi i}\int_{\delta D} g\frac{df}{f} = \sum_{s \in D} g(z) - \sum_{p \in D}g(p)

where {s} and {p} run over the zeroes and poles of {f} in {D}, respectively. In particular, taking {g(z)=z}, we see that

\displaystyle \frac{1}{2\pi i} \int_{\delta D} z\frac{f'(z)dz}{f(z)} = \sum_{z \in D}v_z(f) z

where {v_p(f)} denotes the order of {f} at {z}, i.e. the greatest integer {n} such that {(z-p)^{-n}f(z)} is holomorphic at {z}.

Now if {f} is an elliptic function, and we take for {D} a fundamental parallelogram, we see immediately using the periodicity of {f}, that {\frac{1}{2\pi i} \int_{\delta D}z\frac{f'(z)dz}{f(z)}} equals a {\mathbb{Z}}-linear combination of elements of {\Omega}. Thus we see that , for an elliptic function {f},

\displaystyle \sum_{z \in \mathbb{C}/\Omega}v_z(f) z \equiv 0 \mod \Omega.

In particular, if {f} is an elliptic function of order {3}, then we can determine the position of any zero of {f} from knowledge of the position of the other two; the zeroes are always {z}, {Z}, and {-z-Z} (mod {\Omega}).

Now apply this to the elliptic function

\displaystyle F(u)=\wp'(u)-\wp'(z)=\lambda (\wp(u)-\wp(z)),

which is of order {3}. By construction, it has zeroes at {u=z} and at {u=Z}; thus its third zero is at {u=-z-Z}. This means precisely that the line passing through {(\wp(z), \wp'(z))} and {(\wp(Z), \wp'(Z))} also passes through {(\wp(-z-Z), \wp'(-z-Z)) = (\wp(z+Z), -\wp'(z+Z))}. Of course, these three points lie on the cubic {y^2=4x^3-g_2x-g_3}. This explains precisely where the group law comes from. It also shows why we must reflect the third point of intersection of the line through {P} and {Q} across the {x}-axis.

All of this discussion can be carried out in a quite abstract setting using the Riemann-Roch theorem, which allows us to endow any smooth, genus {1} algebraic variety having at least one point with a group structure, as above. It follows from the general construction that the group structure on an elliptic curve {E} is isomorphic to {\mbox{Pic}_0(E)}, the degree {0} Picard group of {E}.

In a future series of posts, I will discuss the similarities between the theory of number fields and the theory of elliptic curves, which lead to the Birch and Swinnerton-Dyer conjecture.