Pseudo-exponential maps, variants, and quasiminimality

Bays, Martin; Kirby, Jonathan

doi:10.2140/ant.2018.12.493

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Pseudo-exponential maps, variants, and quasiminimality

Martin Bays and Jonathan Kirby

Vol. 12 (2018), No. 3, 493–549

DOI: 10.2140/ant.2018.12.493

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- $℘$ -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.

Keywords

exponential fields, predimension, categoricity, Schanuel conjecture, Ax–Schanuel, Zilber–Pink, quasiminimality, Kummer theory

Mathematical Subject Classification 2010

Primary: 03C65

Secondary: 03C75, 12L12

Milestones

Received: 17 March 2017

Revised: 13 November 2017

Accepted: 26 December 2017

Published: 12 June 2018

Authors

Martin Bays
Institut für Mathematische Logik und Grundlagenforschung, Fachbereich Mathematik und Informatik Universität Münster Münster Germany

Jonathan Kirby
School of Mathematics University of East Anglia Norwich United Kingdom

Vol. 12, No. 3, 2018