Pseudo-exponential maps, variants, and quasiminimality

Abstract

We give a construction of quasiminimal fields equipped with pseudo-analytic maps, generalizing 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 if 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 states only that the graph of exponentiation has nonempty intersection with certain algebraic varieties but does not require genericity of any point in the intersection. Furthermore, Schanuel’s conjecture is not required as a condition for quasiminimality.