The Weierstrass $\wp$-function of the hexagonal lattice

TL;DR

This paper explores properties of the Weierstrass -function related to the hexagonal lattice, utilizing classical theorems to identify specific zeros of its derivative, advancing understanding of elliptic functions in this context.

Contribution

It provides new insights into the zeros of the -function's derivative for the hexagonal lattice using classical theorems, enriching elliptic function theory.

Findings

Zeros of '(' ') ext{ are explicitly characterized.

Utilizes Baker's theorem to analyze meromorphic solutions.

Enhances understanding of elliptic functions associated with hexagonal lattices.

Abstract

We present some properties of the Weierstrass $\wp$-function associated to the hexagonal (or triangular) lattice. In particular, with the help of an old theorem of I.N. Baker \cite{B} on the characterization of meromorphic solutions of the equation $X^3 + Y^3 = 1$ we determine the zeros of the function $\wp'(z) \pm \sqrt{3}$.