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 functions $\wp$, $\sigma$, $\zeta$ and many identities about them
• the Eisenstein series $G_n$ (including the “forbidden” series $G_2$)
• the ODE satisfied by $\wp$ and the corresponding recurrence relation for $G_n$
• the lattice invariants $g_2$, $g_3$, Klein's $J$ invariant, and the elliptic modulus $\lambda$
• the non-vanishing of the lattice discriminant $\Delta$
• the quasiperiods $\eta_1$ and $\eta_2$, Legendre's relation, and their connection to $G_2$
• $G_n$, $\Delta$, $J$, $\lambda$ as holomorphic functions in the upper half plane and the Fourier expansions of the first three
• Dedekind's $\eta$ function and its connection to $G_2$ and $\Delta$
• the functional equations of $G_n$, $\Delta$, $J$, and $\eta$ w.r.t. the modular group
• the inversion formulas for the Jacobi theta functions and their connection to all of the above

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.