Natural models of theories of green points

TL;DR

This paper constructs explicit models of theories of green points in complex fields, depending on parameters, and proves their validity under certain conjectures, with some unconditional results in specific cases.

Contribution

It provides explicit expansions of complex fields as models of green point theories, depending on parameters, and establishes their validity under Schanuel's conjecture or similar assumptions.

Findings

Models depend on parameters and satisfy theories under conjectures.

Unconditional results are obtained for generic parameters in the multiplicative case.

The work links green point theories with complex field expansions.

Abstract

We explicitly present expansions of the complex field which are models of the theories of green points in the multiplicative group case and in the case of an elliptic curve without complex multiplication defined over $\mathbb{R}$. In fact, in both cases we give families of structures depending on parameters and prove that they are all models of the theories, provided certain instances of Schanuel's conjecture or an analogous conjecture for the exponential map of the elliptic curve hold. In the multiplicative group case, however, the results are unconditional for generic choices of the parameters.