A Recipe For Obtaining Algebraic Addition Theorems Of The Weierstrass Elliptic Function

TL;DR

This work addresses the problem of obtaining explicit algebraic addition theorems for the Weierstrass elliptic function $\wp$ in terms of prescribed derivatives and constants. It introduces a determinant-based recipe grounded in Frobenius–Stickelberger identities: by solving a linear system that equates linear combinations of $\wp^{(n_i)}$ and $\wp^{(k_i)}$ at the summands, one constructs a polynomial $\varphi_{n,k,\gamma,\lambda}(s)$ whose zeros lead to an addition relation for $\wp(z_1+\cdots+z_\ell)$. The paper develops a general framework (and its explicit computational steps) to derive classical and novel addition formulas, including two-term, three-term, duplication, and triplication formulas, as well as new identities for the elliptic invariants $g_2$ and $g_3$, all while avoiding conjugate-variable techniques and requiring no lattice symmetry assumptions. The approach yields determinant-based identities and conversion to expressions solely in $\wp$ and its derivatives via standard differential relations, providing a flexible, algorithmic tool for elliptic-function identities with potential generalizations to higher genus. This advances explicit, parameter-rich addition theorems and offers practical formulas for computational and theoretical applications in elliptic function theory.

Abstract

In this paper, we present a general method for obtaining addition theorems of the Weierstrass elliptic function $\wp(z)$ in terms of given parameters. We obtain the classical addition theorem for the Weierstrass elliptic function as a special case. Furthermore, we give novel two-term addition, three-term addition, duplication and triplication formulas. New identities for elliptic invariants are also proven.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Theorem 2.5