Neville's primitive elliptic functions: the case ${\bf g_3 = 0}$

TL;DR

This paper investigates Neville's primitive elliptic functions specifically in the case where the invariant g_3 equals zero, revealing unique properties related to the associated Weierstrass function.

Contribution

It characterizes the structure of Neville's primitive elliptic functions when g_3=0, highlighting the connection to midpoint lattices and square roots of Weierstrass functions.

Findings

g_3=0 implies a midpoint lattice structure

The Weierstrass function admits a square-root representation

Special properties emerge in Neville's functions for g_3=0

Abstract

The vanishing of the invariant $g_3$ attached to a lattice $\Lambda$ singles out a midpoint lattice and yields a square-root of the associated Weierstrass function $\wp_{\Lambda}$.