Pfaffian definitions of Weierstrass elliptic functions

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 $$\sigma $$ . Our work has immediate applications to Diophantine geometry and we answer a question of Corvaja, Masser and Zannier on additive extensions of elliptic curves. We also point out further applications, also in connection with Pila–Wilkie type counting problems.