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--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.
