Complex Lattices, Elliptic Functions, and the Modular Group

This entry defines complex lattices, i.e. $\Lambda(\omega_1,\omega_2) = \mathbb{Z}\omega_1 + \mathbb{Z}\omega_2$ where $\omega_1/\omega_2\notin\mathbb{R}$. Based on this, various other related topics are covered:

• the modular group $\Gamma$ and its fundamental region
• elliptic functions and their basic properties
• the Weierstraß elliptic function $\wp$ and the fact that every elliptic function can be written in terms of $\wp$
• the Eisenstein series $G_n$ (including the forbidden series $G_2$)
• the ordinary differential equation satisfied by $\wp$, the recurrence relation for $G_n$, and the addition and duplication theorems for $\wp$
• the lattice invariants $g_2$, $g_3$, and Klein's $J$ invariant
• the non-vanishing of the lattice discriminant $\Delta$
• $G_n$, $\Delta$, $J$ as holomorphic functions in the upper half plane
• the Fourier expansion of $G_n(z)$ for $z\to i\infty$
• the functional equations of $G_n$, $\Delta$, $J$, and $\eta$ w.r.t. the modular group
• Dedekind's $\eta$ function
• the inversion formulas for the Jacobi $\theta$ functions

In particular, this entry contains most of Chapters 1 and 3 from Apostol's Modular Functions and Dirichlet Series in Number Theory and parts of Chapter 2.

The purpose of this entry is to provide a foundation for further formalisation of modular functions and modular forms.