Recursive spectra of strongly minimal theories satisfying the Zilber trichotomy

TL;DR

This paper explores the recursive presentability of models in strongly minimal theories satisfying the Zilber Trichotomy, proposing a conjecture with partial proofs for specific classes of theories.

Contribution

It formulates a conjecture on the recursive spectra of such theories and proves it for disintegrated theories and modular groups, extending known results for fields.

Findings

Confirmed the conjecture for disintegrated theories

Confirmed the conjecture for modular groups

Known results support the conjecture for fields

Abstract

We conjecture that for a strongly minimal theory T in a finite signature satisfying the Zilber Trichotomy, there are only three possibilities for the recursive spectrum of T: all countable models of T are recursively presentable; none of them are recursively presentable; or only the zero-dimensional model of T is recursively presentable. We prove this conjecture for disintegrated (formerly, trivial) theories and for modular groups. The conjecture also holds via known results for fields. The conjecture remains open for finite covers of groups and fields.