Complex Analysis

4 Elliptic functions

Given a lattice \(\Gamma \) in \(\CC \), we can consider the Riemann surface (elliptic curve) given by \(E=\CC /\Gamma \). \(dz\) gives a trivialization of the canonical bundle of \(E\), and hence of all of its powers. Thus to give a meromorphic function or differential form it suffices to give a \(\Gamma \)-periodic function \(f\), called an elliptic function. Let \(\CC (E)\) be the field of elliptic functions. Given an elliptic function \(f\), let \(v_p(f)\) be the order of the zero/pole at \(p\). For a fixed nonzero \(f\) only finitely many of these can be nonzero. The facts marked RS denote that the theorem holds for some more general type of Riemann surface.

Lemma 4.1 (RS). \(\sum _p v_p(f) = 0\).

Proof. Use the argument principle and integrate around a fundamental domain. One one hand the sides cancel out so we get \(0\), and on the other hand, we get the sum of the orders of the poles and zeroes on the interior. □

The order of an elliptic function is the sum of the orders of the zeros.

Corollary 4.2 (RS). A holomorphic elliptic function is constant.

Proof. This follows from Louisville’s theorem and the previous lemma. □

Lemma 4.3 (RS). The sum of the residues of an elliptic function is \(0\).

Proof. Integrate \(f\) around the fundamental domain. □

Corollary 4.4 (RS). An elliptic function cannot have a unique simple pole.

Proof. The sum of the residues would be nonzero. □

Now we let \(\HH \) be the upper half plane, and consider \(M\), the quotient of \(\HH /\PSL _2(\ZZ )\) via fractional linear transformations. This space classifies lattices up to homothety, and hence elliptic curves, by identifying the lattice \([\tau ,1]\) with \(\tau \), where \(\tau \in \HH \). Note that \(d(\frac {az+b}{cz+d}) = \frac {1}{(cz+d)^2}dz\), so that a section of \(\omega ^k\) on \(\HH /\PSL _2(\ZZ )\) is given by a weakly modular function of weight \(2k\). Really, we want to compactify \(\HH \) to include the point \(\PP ^1_\QQ \), so that a section of \(\omega ^k\) is a modular form of weight \(k\).

We will construct a function on the universal cover of the universal elliptic curve as follows: \(\wp _\tau (z) = \frac {1}{z^2} + \sum '_{w \in \Gamma _\tau }((z+w)^{-2}-w^{-2})\), where the sum is over nonzero lattice points. It isn’t hard to see that this converges uniformly on set bounded away from lattice points, so defines a meromorphic function. To see this is indeed elliptic, first note its derivative \(\frac {\partial \wp }{\partial z} = \wp ' = \sum _{w \in \Gamma _\tau } \frac {-2}{(z+w)^{3}}\) is clearly odd, elliptic, and is order \(3\) since it has a pole at the lattice points. Thus after translation by a fixed lattice point, \(\wp \) can change by some constant. But \(\wp \) is even, so that constant had better be \(0\), and so it is elliptic. It is easy to see that the zeroes of \(\wp '\) are two torsion, since the opposite terms in the sum cancel out, and these must be simple as \(\wp '\) is order \(3\).

Proposition 4.5. An even elliptic function is a rational function in \(\wp \).

Proof. Let \(f\) be our function. If \(f\) has a pole at a non-lattice point \(c\), we can multiply \(f\) by a power of \(\wp (z)-\wp (c)\) to remove it. We are left with a function that only has poles at the lattice point. Note that if \(f\) has a zero at a point of \(2\)-torsion, it must have a double zero, as it is even with respect to that point as well. Thus we can subtract a constant from \(f\) until it has a zero in common with \(\wp \), and then divide by \(\wp \). This lowers the order, as it will cancel out zeroes and poles, so repeat this process. □

Corollary 4.6. \(\CC (E) = \CC (\wp ,\wp ')\) where \(\wp '^2 = 4\prod _{x \in E[2]}(\wp (z)-\wp (x))\).

Proof. Every function is the sum of an even and odd function. An odd function times \(\wp '\) is even, and so we get the first part. For the second part, note that the two functions have the same zeroes and poles, and the factor of \(4\) comes from looking at the expansion around \(0\). □

By looking at the series expansion at the origin, one sees that \(\pp _\tau (z) = \pp _{\tau +1}(z)\). Moreover, \(\pp _{\frac {-1}{\tau }}(z) = \tau ^2\pp _\tau (\tau z)\).

Given a lattice \([b,a]\) , define \(G_k\), \(k\geq 1\) to be the sum \(\sum _{m \in \ZZ } \sum '_{n \in \ZZ }\frac {1}{(na+mb)^{2k}}\) where the sum runs over the nonzero lattice points. Because of the way that \(G_k\) scales, it is weakly modular of weight \(k\) when \(k\geq 2\) When \(k = 1\) it is still invariant under \(\tau \mapsto \tau +1\). And indeed, we can compute the \(q\)-expansion, so these are actually modular for \(\PSL _2(\ZZ )\) when \(k>1\). First, \(\pi (1-2\sum _0^\infty q^n) = \pi \frac {q+1}{q-1} = \pi \cot (\pi \tau ) = \sum \frac {1}{\tau +m}\), and so taking the \(k-1^{th}\) derivative (\(k\geq 2\)), we get \(\sum \frac {(-2\pi i)^k}{(k-1)!}\sum _{l\geq 0} l^{k-1}q^l = \sum \frac {1}{({\tau +m})^k}\). Thus \(G_k(\tau ) = 2\zeta (2k) + 2\sum _1^\infty \sum _{\ZZ }\frac {1}{(n\tau +m)^{2k}} = 2\zeta (2k) + \frac {2(-1)^k(2\pi )^{2k}}{(2k-1)!}\sum _1^{\infty }\sigma _{2k-1}(l)q^l\).

Moreover, looking at the definition of \(\wp \), we see that \(\wp (z) = \frac 1 {z^2} + \sum _1^\infty (2k+1)G_{k+1}z^{2k}\). Moreover, comparing the negative terms of \(\wp ,(\wp ')^2,\wp ^3\) near \(0\), we get that \(\wp '^2 = 4\wp ^3-g_2\wp -g_3\) where \(g_2 = 60G_2\), \(g_3 = 140G_3\).