Pfaffian definitions of Weierstrass elliptic functions

Abstract

We give explicit definitions of the Weierstrass elliptic functions $\wp$ and $ζ$ 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 $σ$. 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.

38 pages. Fixed a typo, and slightly expanded the introduction. Comments welcome!