A nondefinability result for expansions of the ordered real field by the Weierstrass $\wp$ function

TL;DR

This paper characterizes when the Weierstrass  function restricted to a disc is definable in a structure based on the real field and the function on an interval, linking it to complex multiplication of lattices.

Contribution

It provides a nondefinability result for expansions of the real field by the  function, characterizing lattices with complex multiplication through definability.

Findings

 restricted to a disc is definable iff the lattice has complex multiplication.

The structure  on an interval determines the lattice's complex multiplication property.

The result links complex multiplication to logical definability in real-analytic structures.

Abstract

Suppose that $\Omega$ is a complex lattice that is closed under complex conjugation and that $I$ is a small real interval, and that $D$ is a disc in $ \mathbb{C}$. Then the restriction $\wp|_D$ is definable in the structure $(\bar{\mathbb{R}},\wp|_I)$ if and only if the lattice $\Omega$ has complex multiplication. This characterises lattices with complex multiplication in terms of definability.