Elliptic Functions and Modular Forms

A function that is doubly periodic — invariant under a lattice of translations — is the natural meromorphic function on a torus, and its theory closes the classical loop between complex analysis, Riemann surfaces, and number theory. This page develops the Weierstrass $\wp$-function, the cubic it satisfies, theta functions, and the modular group acting on the space of lattices. It builds on complex-plane.md, laurent-residues.md, and infinite-products.md.

Lattices and elliptic functions

Fix a lattice $\Lambda =\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2$ (with ${\omega }_2/{\omega }_1\notin \mathbb{R}$). A meromorphic $f$ is elliptic if $f(z+\omega )=f(z)$ for all $\omega \in \Lambda$; equivalently it is a meromorphic function on the torus $\mathbb{C}/\Lambda$ — a compact Riemann surface of genus $1$. Integrating over a fundamental parallelogram (the residue theorem with cancelling opposite sides) gives Liouville's theorems for elliptic functions:

Liouville's theorems. A holomorphic elliptic function is constant (bounded entire on the torus); the sum of residues in a period parallelogram is $0$; the number of poles equals the number of zeros (the order, always $\ge 2$); and the sum of the zeros equals the sum of the poles mod $\Lambda$.

So there is no elliptic analogue of $z$ or $e^z$: the simplest nonconstant elliptic functions have order $2$.

The Weierstrass $\wp$-function

The order-2 elliptic function with a double pole at each lattice point is built by the Mittag-Leffler recipe: $$\wp (z)=\frac{1}{z^2}+\sum _{\omega \in \Lambda \setminus \{0\}}\left(\frac{1}{(z-\omega {)}^2}-\frac{1}{{\omega }^2}\right).$$ It is even, elliptic, and its derivative ${\wp }^{\prime }$ is odd and elliptic of order $3$. Expanding at $0$ and matching Laurent coefficients yields the central identity:

The Weierstrass cubic. $\wp$ satisfies the differential equation $$({\wp }^{\prime }{)}^2=4{\wp }^3-g_2\wp -g_3,$$ where $g_2=60{\sum }^{\prime }{\omega }^{-4}$ and $g_3=140{\sum }^{\prime }{\omega }^{-6}$ are the Eisenstein series of the lattice.

Thus $z\mapsto (\wp (z),{\wp }^{\prime }(z))$ maps the torus $\mathbb{C}/\Lambda$ isomorphically onto the elliptic curve $y^2=4x^3-g_{2}x-g_3$ in the projective plane — the analytic uniformization of a genus-1 curve, and the reason complex analysis, algebraic geometry, and the arithmetic of elliptic curves share a subject. The group law on the curve is just addition on $\mathbb{C}/\Lambda$.

Theta functions

An alternative construction assembles elliptic functions from theta functions — entire functions quasi-periodic under $\Lambda$: $$\theta (z\mid \tau )=\sum _{n\in \mathbb{Z}}{e}^{\pi i{n}^2\tau +2\pi inz},\tau ={\omega }_2/{\omega }_1,\mathrm{Im}\tau >0.$$ Ratios of theta functions are elliptic, and $\wp$ is a second logarithmic derivative of $\theta$. Theta's modular transformation $\theta (1/t)=\sqrt{t}\theta (t)$ is the Poisson-summation identity that drives the functional equation of the zeta function.

The modular group and modular forms

An elliptic function depends only on the lattice, and two lattices give isomorphic tori iff their ratios $\tau$ are related by the modular group $$PSL(2,\mathbb{Z}):\tau \mapsto \frac{a\tau +b}{c\tau +d},\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in SL(2,\mathbb{Z}),$$ a discrete subgroup of the Möbius group acting on the upper half-plane $\mathbb{H}$. Functions on lattices transforming with a fixed weight under this action are modular forms:

Definition. A modular form of weight $k$ is a holomorphic $f$ on $\mathbb{H}$ with $f\left(\frac{a\tau +b}{c\tau +d}\right)=(c\tau +d{)}^{k}f(\tau )$ and bounded as $\mathrm{Im}\tau \to \infty$. The Eisenstein series $G_k$ are the basic examples.

The absolute invariant is the $j$-invariant $$j(\tau )=1728\frac{g_2^3}{g_2^3-27g_3^2},$$ a modular function ($k=0$) that is a bijection from the moduli space $\mathbb{H}/PSL(2,\mathbb{Z})$ onto $\mathbb{C}$: two tori are isomorphic iff they share a $j$-value. The closely related modular $\lambda$-function is the covering map $\mathbb{H}\to \mathbb{C}\setminus \{0,1\}$ behind the Picard theorems.

References

• Ahlfors, Complex Analysis, Ch. 7 (elliptic and modular functions).
• Stein & Shakarchi, Complex Analysis, Ch. 9 (elliptic functions, theta).
• Serre, A Course in Arithmetic, Ch. VII (modular forms).