Pfaffian definitions of Weierstrass elliptic functions

TL;DR

This paper provides explicit pfaffian-based definitions of Weierstrass elliptic functions with lattice-independent complexity, enabling uniform zero estimates and addressing questions on elliptic curve extensions.

Contribution

It introduces novel pfaffian definitions for Weierstrass functions that are independent of the lattice, facilitating new applications in elliptic curve theory.

Findings

Explicit pfaffian definitions with lattice-independent complexity

Uniform zero estimates for the Weierstrass $ ext{wp}$ function

Resolution of a question on additive extensions of elliptic curves

Abstract

We give explicit definitions of the Weierstrass elliptic functions $\wp$ and $\zeta$ in terms of pfaffian functions, with complexity independent of the lattice involved. We also give such a definition for a modification of the Weierstrass function $\sigma$. As immediate applications, we give an explicit uniform zero estimate for $\wp$ and answer a question of Corvaja, Masser and Zannier on additive extensions of elliptic curves.