Pseudo-exponential maps, variants, and quasiminimality

TL;DR

This paper constructs quasiminimal fields with pseudo-analytic maps, extending Zilber's pseudo-exponential, and explores conditions under which the complex exponential field is quasiminimal, relating to Schanuel's conjecture.

Contribution

It introduces a new construction of quasiminimal fields with pseudo-analytic maps, including pseudo-$p$-functions for elliptic curves, and analyzes their relation to Schanuel's conjecture and exponential-algebraic closedness.

Findings

Constructs pseudo-exponential maps for simple abelian varieties.

Shows the complex field with these maps is isomorphic to the pseudo-analytic version under certain conjectures.

Demonstrates the complex exponential field is quasiminimal if exponentially-algebraically closed.

Abstract

We give a construction of quasiminimal fields equipped with pseudo-analytic maps, generalising Zilber's pseudo-exponential function. In particular we construct pseudo-exponential maps of simple abelian varieties, including pseudo-$\wp$-functions for elliptic curves. We show that the complex field with the corresponding analytic function is isomorphic to the pseudo-analytic version if and only the appropriate version of Schanuel's conjecture is true and the corresponding version of the strong exponential-algebraic closedness property holds. Moreover, we relativize the construction to build a model over a fairly arbitrary countable subfield and deduce that the complex exponential field is quasiminimal if it is exponentially-algebraically closed. This property asks only that the graph of exponentiation have non-trivial intersection with certain algebraic varieties but does not require genericity of these points. Furthermore Schanuel's conjecture is not required as a condition for quasiminimality.