Abstract
We give a construction of quasiminimal fields equipped with pseudoanalytic maps, generalizing Zilber’s pseudoexponential function. In particular we construct pseudoexponential 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 pseudoanalytic version if and only if the appropriate version of Schanuel’s conjecture is true and the corresponding version of the strong exponentialalgebraic 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 exponentiallyalgebraically 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.
℘
functions for elliptic curves. We show that the complex field with the corresponding analytic function is isomorphic to the pseudoanalytic version if and only if the appropriate version of Schanuel’s conjecture is true and the corresponding version of the strong exponentialalgebraic 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 exponentiallyalgebraically 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.
Original language  English 

Pages (fromto)  493–549 
Journal  Algebra and Number Theory 
Volume  12 
Issue number  3 
DOIs  
Publication status  Published  12 Jun 2018 
Keywords
 exponential fields
 predimension
 categoricity
 Schanuel conjecture
 Ax–Schanuel
 Zilber–Pink
 quasiminimality
 Kummer theory
Profiles

Jonathan Kirby
 School of Mathematics  Reader
 Logic  Member
Person: Research Group Member, Academic, Teaching & Research