Real geometric transcendence for Weierstrass elliptic functions

TL;DR

This paper characterizes algebraic curves in the plane whose images under Weierstrass elliptic functions are contained in algebraic curves, advancing understanding of the geometric transcendence properties of these functions.

Contribution

It provides a classification of algebraic curves in the plane related to Weierstrass elliptic functions, revealing new geometric transcendence results.

Findings

Identifies algebraic curves mapped into algebraic curves by Weierstrass functions

Establishes conditions for algebraic curve images under elliptic functions

Advances the understanding of geometric transcendence in elliptic functions

Abstract

Given a lattice $\Lambda \subset \mathbb C\simeq \mathbb R^2$ with associated Weierstrass function $\wp_{\Lambda}$, we determine the algebraic curves in $\mathbb R^2$ whose image via $\wp_{\Lambda}$ is contained in an algebraic curve.