Complex Lattices, Elliptic Functions, and the Modular Group

Manuel Eberl, Anthony Bordg, Wenda Li, Lawrence C. Paulson

Abstract: 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.

External IDs:dblp:journals/afp/EberlBLP25