An introduction to Elliptic Functions

(section 9) An introduction to Elliptic Functions

(subsection 1) Elliptic functions

(Def) meromorphic functions f on \mathbb{C} that have two periods; there are two non-zaro complex numbers \omega_{1} and \omega_{2} s.t. f(z + w_{1}) = f(z) and f(z+w_{2}) = f(z) for all z \in \mathbb{C} . A function with two periods is said to be doubly periodic.

Assume f is a meromorphic function on \mathbb{C} with periods 1 and \tau where Im(\tau) >0. Successive applications of the periodicity conditions yield f(z + n + m \tau) = f(z) for all integers n,m and all z \in \mathbb{C}, and it is therefore natural to consider the lattice in \mathbb{C} defined by \Lambda = {n + m\tau : n , m \in \mathbb{Z}} . We say that 1 and \tau generate \Lambda.

Associated to the lattice \Lambda is the fundamental parallelogram defined by P_{0} = {z \in \mathbb{C} : z = a + b \tau where 0 \le a < 1 and 0 \le b < 1}.

(Def) Two complex numbers z and w are congruent modulo \Lambda if z = w + n + m \tau for some n , m \in \mathbb{Z}, and we write z \sim w. in other words, z and w differ by a point in the lattice, z – w \in \Lambda.

(Def) A period parallelogram P is any translate of the fundamental parallelogram , P = P_{0} + h with h \in \mathbb{C}.

(Prop) \Lambda and P_{0} give rise to a covering (or tiling) of the complex plane \mathbb{C} = \bigcup_{n , m \in \mathbb{Z}} (n + m\tau + P_{0}), and this union is disjoint.

(Prop 1.1.) Suppose f is a meromorphic function with two periods 1 and \tau which generate the lattice \Lambda. Then :

Every point in \mathbb{C} is congruent to a unique point in the fundamental parallelogram.

Every point in \mathbb{C} is congruent to a unique point in the fundamental parallelogram.

The lattice \Lambda provides a disjoint covering of the complex plane, in the sense of above (prop).

The function f is completely determined by its values in any period parallelograms.

(subsubsection 1.1.) Liouville’s theorems

(Thm 1.2.) An entire doubly periodic function is constant.

(Def) A non-constant doubly periodic meromorphic function is called an elliptic function.

(Thm 1.3.) The total number of poles of an elliptic function in P_{0} is always \ge 2.

(Thm 1.4.) Every elliptic function of order m has m zeros in P_{0}.

(subsubsection 1.2.) The Weierstrass \varphi function

(Def) Let \Lambda^{} denote the lattice minus the origin, that is, \Lambda^{} = \Lambda – {(0,0)}, and consider instead the following series : \frac{1}{z^{2}} + \sum_{\omega \in \Lambda^{*}} [\frac{1}{(z+\omega)^{2}} - \frac{1}{\omega^{2}}].

(Lem 1.5.) The two series \sum_{(n,m) \neq (0,0)} \frac{1}{(|n| + |m|)^{r}} and \sum_{n+m \tau \in \Lambda^{*}} \frac{1}{|n+m\tau|^{r}} converge if r > 2.

(Def) Define Weierstrass \varphi function, which is given by the series \frac{1}{z^{2}} + \sum_{\omega \in \Lambda^{*}} [\frac{1}{(z+\omega)^{2}} - \frac{1}{\omega^{2}}]. = \frac{1}{z^{2}} + \sum_{(n,m) \neq (0,0)}[\frac{1}{(z + n + m \tau}^{2} - \frac{1}{(n+m\tau)^{2}}].

\varphi is a meromorphic function with double poles at the lattice points.

(Thm 1.6.) The function \varphi is an elliptic function that has periods 1 and \tau, and double poles at the lattice points.

(Def) Since \varphi’ is elliptic and has order 3, the three points 1/2, \tau/2, and (1+\tau)/2 (which are called the half-periods) are the only roots of \varphi’ in the fundamental parallelogram, and they have multiplicity 1. Therefore, if we define \varphi(1/2) = e_{1}, \varphi(\tau/2) = e_{2} and \varphi(\frac{1+\tau}{2}) = e_{3}

(Thm 1.7.) The function (\varphi’)^{2} is the cubic polynomial in \varphi (\varphi’)^{2} = 4(\varphi – e_{1}) (\varphi -e_{2}) (\varphi – e_{3}) .

(Thm 1.8.) Every elliptic function f with periods 1 and \tau is a rational function of \varphi and \varphi ‘.

(Lem 1.9.) Every even elliptic function F with periods 1 and \tau is a rational function of \varphi.

(subsection 2) The modular character of elliptic functions and Eisenstein series

(Def) Consider the group of transformations of the upper half-plane Im(\tau) >0, generated by the two transformations \tau \mapsto \tau +1 and \tau \mapsto -1/\tau. This group is called the modular group.

(subsubsection 2.1.) Eisenstein series

(Def) The Eisenstein series of order k is defined by E_{k}(\tau) = \sum_{(n,m) \neq (0,0)} \frac{1}{(n+m\tau)^{k}}, whenever k is an integer \ge 3 and \tau is a complex number with Im(\tau) >0. If \Lambda is the lattice generated by 1 and \tau, and if we write \omega = n + m \tau, then another expression for the Eisenstein series is \sum_{\omega \in \Lambda^{*}} 1/\omega^{k}.

(Thm 2.1.) Eisenstein series have the following properties :

The series E_{k}(\tau) converges if k \ge 3, and is holomorphic in the upper half-plane.

E_{k}(\tau) = 0 if k is odd.

E_{k}(\tau) satisfies the following transformation relations:

E_{k}(\tau +1) = E_{k}(\tau) and E_{k}(\tau) = \tau^{-k} E_{k}(-1/\tau).

The last property is sometimes referred to as the modular character of the Eisenstein series.

(Thm 2.2.) For z near 0, we have \varphi(z) = \frac{1}{z^{2}} + 3E_{4}z^{2} + 5E_{6}z^{4} + \cdot = \frac{1}{z^{2}} + \sum_{k=1}^{\infty} (2k+1) E_{2k+2} z^{2k}.

(Cor 2.3.) If g_{2} = 60E_{4} and g_{3} = 140E_{6}, then (\varphi ‘)^{2} = 4 \varphi^{3} – g_{2} \varphi – g_{3}.

(subsubsection 2.2.) Eisenstein series and divisor functions

(Lem 2.4.) If k \ge 2 and Im(\tau) >0, then \sum_{n=-\infty}^{\infty} \frac{1}{(n+\tau)^{k}} = \frac{(-2\pi i)^{k}}{(k-1)!} \sum_{l=1}^{\infty} l^{k-1} e^{2\pi i \tau l}.

(Def) The divisor function \sigma_{l}(r) that arises here is defines as the sum of the l^{th} powers of the divisors of r, that is , \sigma_{l}(r) = \sum_{d|r} d^{l} .

(Thm 2.5.) If k \ge 4 is even, and Im(\tau) >0, then E_{k}(\tau) = 2 \zeta(k) + \frac{2(-1)^{k/2} (2\pi)^{k}}{(k-1)!}\sum_{r=1}^{\infty} \sigma_{k-1} (r) e^{2\pi i \tau r}.

(Cor 2.6.) The double sum defining F converges in the indicated order. We have F(\tau) = 2 \zeta(2) – 8 \pi^{2} \sum_{r=1}^{\infty} \sigma(r) e^{2\pi i r \tau}, where \sigma(r) = \sum_{d|r} d is the sum of the divisors of r.