Nondefinability results for elliptic and modular functions

Abstract

Let $\Omega$ be a complex lattice which does not have complex multiplication and $\wp ={\wp }_{\Omega }$ the Weierstrass $\wp$-function associated to it. Let $D\subseteq \mathbb{C}$ be a disc and $I\subseteq \mathbb{R}$ be a bounded closed interval such that $I\cap \Omega =\varnothing$. Let $f:D\to \mathbb{C}$ be a function definable in $(\overline{\mathbb{R}},\wp {|}_I)$. We show that if $f$ is holomorphic on $D$ then $f$ is definable in $\overline{\mathbb{R}}$. The proof of this result is an adaptation of the proof of Bianconi for the ${\mathbb{R}}_{\exp }$ case. We also give a characterization of lattices with complex multiplication in terms of definability and a nondefinability result for the modular $j$-function using similar methods.

17 pages, comments welcome. Small change in abstract, content otherwise unchanged