Elliptic functions

I am writing this as a way to go through in detail the section on elliptic functions in Schlag’s book.

Proposition 4.14.  Let \Lambda = \{m\omega_1 + n\omega_2 | m,n \in \mathbb{Z}\} and set \Lambda^* = \Lambda \setminus \{0\}. For any integer n \geq 3, the series

f(z) = \displaystyle\sum_{w \in \Lambda} (z+w)^{-n} \qquad (4.16)

defines a function f \in \mathcal{M}(M) with deg(f) = n. Furthermore, the Weierstrass function

\wp(z) = \frac{1}{z^2} + \displaystyle\sum_{w \in \Lambda^*} [(z+w)^{-2} - w^{-2}] ,\qquad (4.17)

is an even elliptic function of degree two with \Lambda as its group of periods. The poles of \wp are precisely the points in \Lambda and they are all of order 2.

Proof.  It suffices to prove that f(z) = \displaystyle\sum_{w \in \Lambda} (z+w)^{-n} converges absolutely and uniformly on every compact set K \subset \mathbb{C} \setminus \Lambda. Periodicity allows us to restrict to the closure of any fundamental region. There exists C > 0 such that for all x,y \in \mathbb{R},

C^{-1}(|x|+|y|) \leq |x\omega_1 + y\omega_2|.

Hence, when z \in \{x\omega_1 + y\omega_2 | 0 \leq x, y \leq 1\}, then

|z + (k_1\omega_1 + k_2\omega_2)| \geq C^{-1}(|k_1| + |k_2|) - |z| \geq (2C)^{-1}(|k_1| + |k_2|)

provided |k_1| + |k_2| is sufficiently large. In

\displaystyle\sum_{|k_1|+|k_2|>0} |k_1 + k_2|^{-n},

there are O(n) occurrences of |k_1| + |k_2| = n, which means the above converges when n > 2, and this, with the above bound, means f \in \mathcal{H}(\mathbb{C} \setminus \Lambda). Periodicity implies f \in \mathcal{M}(M). Moreover, the degree of (4.16) is determined by nothing that inside a fundamental region the series has a unique pole of order n.

For the second part, we note that when |w| > 2|z|,

\left|(z+w)^{-2} - w^{-2}\right| \leq \frac{|z||z+2s|}{|w|^2|z+w|^2} \leq \frac{C|z|}{|w|^3},

which means the series defining \wp, which is clearly even, converges absolutely and uniformly on compact subsets of \mathbb{C} \setminus \Lambda. For the periodicity of \wp, note that \rho' is periodic relative the same lattice \Lambda. Thus, for every w \in \Lambda,

\wp(z+w) - \wp(z) = C(w) \quad \forall z \in \mathbb{C}

with some constant C(w), which has to be zero by

\wp(\omega_1/2) - \wp(-\omega_1/2) = 0.

Another way to go about it to define \sigma such that

\zeta(z) = \frac{d \log \sigma(z)}{dz} = \frac{1}{z} + \displaystyle\sum_{w \in \Lambda^*} \left[\frac{1}{z-\omega} + \frac{1}{\omega} + \frac{z}{\omega^2}\right],

so that \wp = -\zeta', from which by periodicity, we have

\zeta(z+\omega) - \zeta(z) = C(\omega).

From this we can solve that

\sigma(z+\omega_j) = -\sigma(z)e^{\eta_j(z+\omega_j/2)}, \qquad (4.20)

where the \nu_js are constants for j = 1,2.

Lemma 4.15.  With \wp as before, one has

(\wp'(z))^2 = 4(\wp(z) - e_1)(\wp(z) - e_2)(\wp(z) - e_3) \qquad (4.21)

where e_1 = \wp(\omega_1/2), e_2 = \rho(\omega_2/2), and e_3 = \rho((\omega_1+\omega_2)/2) are pairwise distinct. Furthermore, one has e_1 + e_2 + e_3 = 0 so that (4.21) can be written in the form

(\wp'(z))^2 = 4(\wp(z))^3 - g_2\wp(z) - g_3 \qquad (4.22)

with constants g_2 = -4(e_1e_2 + e_1e_3 + e_2e_3) and g_3 = 4e_1e_2e_3.

View the torus as

S = \{x\omega_1 + y\omega_2 | -1/2 \leq x,y \leq 1/2\}.

\wp'(z) is odd and has a pole of order 3 at z = 0 but no other poles in S, which means \wp'(z) has degree 3.

Oddness with periodicity applied at \omega_1/2 and \omega_2/2 yields that

\frac{1}{2}\omega_1, \quad \frac{1}{2}\omega_2, \quad \frac{1}{2}(\omega_1+\omega_2)

are the three zeros of \wp', each simple, and thus also the unique points where \wp has valency 2 apart from z = 0. The e_j are distinct, because if not \wp would assume such a value four times, impossible when the degree is 2.

Denoting the RHS of (4.21) by F(z), we have that

\frac{(\wp'(z))^2}{F(z)} \in \mathcal{H}(M)

with the zeros cancelled out, and thus equal to a constant.

At z = 0, the highest pole of (\wp'(z))^2 is one of order 3\cdot 2 = 6 with coefficient (-2)^2 = 4. In F(z), we have essentially a cubic in \wp(z) with leading coefficient 4, and \wp(z) has pole of order 2 with coefficient 1. In taking the limit towards zero, we only need to consider the 4\wp(z)^3 term, which has the highest order pole, which is also of order 6 with coefficient 4. That means our constant function is 1.

The final statement follows by observing from the Laurent series around zero, which is, from the geometric series of \left(\frac{1/w}{1+(z/w)}\right)^2 - \frac{1}{w^2}

\wp(z) = \frac{1}{z^2} - \displaystyle\sum_{k = 1}^{\infty} (k+1)(-1)^{k}z^k\displaystyle\sum_{w \in \Lambda^*} \frac{1}{w^{3+k}}.

Because \wp is even, the odd coefficients must vanish. So we have

\wp(z) = \frac{1}{z^2} - \displaystyle\sum_{k=1}^{\infty} (2k+1)z^{2k} \displaystyle\sum_{w \in \Lambda^*} \frac{1}{w^{2k+2}}.

For now, let

G_k = \displaystyle\sum_{w \in \Lambda^*} \frac{1}{w^k}.

\begin{aligned}\wp(z) & = & \frac{1}{z^2} + 3G_4z^2 + 5G_6z^4 + \cdots, \\ \wp'(z) & = & \frac{-2}{z^3} + 6G_4z + 20G_6z^3 + \cdots, \\ (\wp(z))^3 & = & \frac{1}{z^6} + 9\frac{G_4}{z^2} + \cdots, \\ (\wp'(z))^2 & = & \frac{4}{z^6} - \frac{24G_4}{z^2} + \cdots. \end{aligned}

What we want is to find the g_2 such that (\wp'(z))^2 - 4(\wp(z))^3 + g_2\wp(z) becomes analytic and thus constant, and to do that we must vanish out all the poles at 0. The z^{-6} coefficients tells us to multiply (\wp(z))^3 by 4. After that, we have from the z^{-2} coefficient that -24G_4 - 9\cdot 4 G_4 + g_2 = 0, which means

g_2 = 60G_4 = -4(e_1e_2 + e_1e_3 + e_2e_3).

Proposition 4.16.  Every f \in \mathcal{M}(M) is a rational function of \wp and \wp'. If f is even, then it is a rational function of \wp alone.

Proof.  Suppose that f is non-constant and even. Then for all but finitely many values of w \in \mathbb{C}_{\infty}, the equation f(z) - w = 0 has only simple zeros (since there are only finitely many zeros of f'). Pick two such w \in \mathbb{C} and denote them by c,d. Moreover, we can ensure that the zeros of f - c and f - d are distinct from the branch points of \wp. Thus, since f is even and with 2n = deg(f), one has:

\begin{aligned}\{z \in M : f(z) - c = 0\} & = \{a_j, -a_j\}_{j=1}^n, \\ \{z \in M : f(z) - d = 0\} & = \{b_j, -b_j\}_{j=1}^n. \end{aligned}

The elliptic functions

g(z) = \frac{f(z) - c}{f(z) - d}

and

h(z) = \displaystyle\prod_{j=1}^n \frac{\wp(z) - \wp(a_j)}{\wp(z) - \wp(b_j)}

have the same zeros and poles which are all simple. It follows that g = \alpha h for some \alpha \neq 0. Solving this relation for f yields the desired conclusion.

If f is odd, then f/\wp' is even so f = \wp'R(\wp) where R is rational. Finally, if f is any elliptic function, then

f(z) = \frac{1}{2}(f(z) + f(-z)) + \frac{1}{2}(f(z) - f(-z))

is a decomposition into even/odd elliptic functions whence

f(z) = R_1(\wp) + \wp'R_2(\wp)

with rational R_1, R_2 as claimed.     ▢

We conclude with the following question: given disjoint finite sets of distinct points \{z_j\} and \{\zeta_k\} in M as well as positive integers n_j for z_j and \nu_k for \zeta_k, respectively, is there an elliptic function with precisely these zeros and poles and of the given orders? In the case of \mathbb{C}_\infty yes iff \sum_{j} n_j = \sum_{k} \nu_k since the degree must be constant throughout.

For the tori, we first observe that by the residue theorem one has

\frac{1}{2\pi i}\oint_{\partial P} z\frac{f'(z)}{f(z)}dz = \sum_j n_jz_j - \sum_k \nu_k \zeta_k. \qquad (4.25)

where \partial P is the boundary of a fundamental region P such that no zero or pole lies on the boundary. Second, comparing parallel sides of the fundamental region and using the periodicity shows that the left-hand side is (4.25) is of the form n_1\omega_1 + n_2\omega_2 with n_1, n_2 \in \mathbb{Z} and thus equals 0 modulo \Lambda. (This follows from that \int_{\gamma} \frac{f'(z)}{f(z)}dz is the difference of logarithms of the same value, which regardless of branch must be an integer multiple of 2\pi i.)

Now consider the edges in \partial P given by \gamma_1(t) = \{t\omega_1 | 0 \leq t \leq 1\} and \gamma_1(t) = \{\omega_2 + t\omega_1 | 0 \leq t \leq 1\} respectively. By \omega_2-periodicity of \frac{f'(z)}{f(z)} we infer that

\int_{\gamma_1} z\frac{f'(z)}{f(z)}dz + \int_{\gamma_2} z \frac{f'(z)}{f(z)}dz = -\omega_2\int_{\gamma_1}d \log f(z).

The branch of logarithm here is irrelevant, since the arbitrary constant is differentiated away. By periodicity applied to the difference in this integral,

\omega_2 \frac{1}{2\pi i} \int_{\gamma_1} d\log f(z) \in \omega_2 \mathbb{Z}.

The other edge pair gives an element of \omega_1 \mathbb{Z}, whence (4.24).

Theorem 4.17.  Suppose (4.23) and (4.24) hold. Then there exists an elliptic function which has precisely these zeros and poles with the given orders. This function is unique up to a nonzero complex multiplicative constant.

Proof.  Listing the points z_j and \zeta_k expanded out with their respective multiplicities, we obtain sequences z_j' and \zeta_k' of the same length, say n. Shifting the z_j's and \zeta_k's by lattice elements if needed, one has

\sum_{j=1}^n z_j' = \sum_{k=1}^n \zeta_k'.

Take

f(z) = \displaystyle\prod_{j=1}^n \frac{\sigma(z - z_j')}{\sigma(z - \zeta_j')}

using the \sigma in (4.20). Then

\begin{aligned} \frac{f(z+\omega_i)}{f(z)} & = & \displaystyle\prod_{j=1}^n \frac{\sigma(z-z_j' + \omega_i/2)}{\sigma(z - z_j')}\cdot \frac{\sigma(z - \zeta_j')}{\sigma(z - \zeta_j' + \omega_i/2)} \\ & = & \displaystyle\prod_{j=1}^n e^{\eta_i\left[(z - z_j' + \omega_i/2) - (z - \zeta_j' + \omega_i/2)\right]} \\ & = & e^{\eta_i\cdot 0} \\ & = & 1, \end{aligned}

which shows periodicity.    ▢

Finally, we observe how we can solve (4.22) by integrating

\frac{d\wp(z)}{\sqrt{4(\wp(z))^3 - g_2\wp(z) - g_3}} = dz

where we choose some branch of the root, which yields

z - z_0 = \int_{\wp(z_0)}^{\wp(z)} \frac{d\zeta}{\sqrt{4\zeta^3 - g_2\zeta - g_3}}.

In other words, the Weierstrass function \wp is the inverse of an elliptic integral. The integration path in (4.30) needs to be chosen to avoid the zeros and poles of \wp', and the branch of the root is determined by \wp'.

Analogously, \int_{w_0}^w \frac{d\zeta}{\sqrt{1 - \zeta^2}} = z - z_0 is satisfied by w = \sin z with similar restriction on the path and the choice of branch. This case though is a periodic function with one period, whereas in (4.30), there are two periods.

References

  • Schlag, W., A Course in Complex Analysis and Riemann Surfaces, American Mathematical Society, 2014, pp. 153-157.

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