Nondefinability results for elliptic and modular functions

TL;DR

The paper investigates the definability of elliptic and modular functions, showing that certain holomorphic functions definable in specific structures are actually definable in the real field, and characterizes lattices with complex multiplication.

Contribution

It extends nondefinability results to elliptic and modular functions, providing new characterizations of lattices with complex multiplication.

Findings

Holomorphic functions definable in certain structures are actually definable in the real field.

Characterization of lattices with complex multiplication via definability.

Nondefinability of the modular j-function in certain structures.

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 $I\cap\Omega=\emptyset$. Let $f:D\rightarrow\mathbb{C}$ 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.