Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T17:09:52.619Z Has data issue: false hasContentIssue false
  • English
  • Français

UNIVERSAL COVERS OF COMMUTATIVE FINITE MORLEY RANK GROUPS

Part of: Model theory Abelian varieties and schemes

Published online by Cambridge University Press:  26 April 2018

Martin Bays ,
Bradd Hart  and
Anand Pillay
Martin Bays
Affiliation:
Institut für Mathematische Logik und Grundlagenforschung, Fachbereich Mathematik und Informatik Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany (mbays@sdf.org)
Bradd Hart
Affiliation:
Department of Mathematics and Statistics McMaster University, 1280 Main St., Hamilton, ON L8S 4K1, Canada (hartb@mcmaster.ca)
Anand Pillay
Affiliation:
Department of Mathematics, University of Notre Dame, 281 Hurley Hall, Notre Dame, IN 46556, USA (apillay@nd.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We give an algebraic description of the structure of the analytic universal cover of a complex abelian variety which suffices to determine the structure up to isomorphism. More generally, we classify the models of theories of ‘universal covers’ of rigid divisible commutative finite Morley rank groups.

Keywords

Abelian varietiesexponential mapKummer theorycategoricityMorley rank

MSC classification

Primary: 03C45: Classification theory, stability and related concepts 03C60: Model-theoretic algebra 14K99: None of the above, but in this section
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 3 , May 2020 , pp. 767 - 799
Copyright
© Cambridge University Press 2018

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bays, M., Categoricity results for exponential maps of 1-dimensional algebraic groups & Schanuel conjectures for powers and the CIT. PhD thesis, Oxford University, 2010.http://wwwmath.uni-muenster.de/u/baysm/dist/thesis/.Google Scholar
Benoist, F., Bouscaren, E. and Pillay, A., Semiabelian varieties over separably closed fields, maximal divisible subgroups, and exact sequences, J. Inst. Math. Jussieu 15(1) (2016), 29–69.Google Scholar
Bertrand, D., Galois descent in Galois theories, in Arithmetic and Galois theories of differential equations, Sémin. Congr., Volume 23 (Soc. Math. France, Paris, 2011), 1–24.Google Scholar
Bays, M., Gavrilovich, M. and Hils, M., Some definability results in abstract Kummer theory, Int. Math. Res. Not. IMRN 2014(14) 3975–4000.Google Scholar
Bays, M., Hart, B., Hyttinen, T., Kesälä, M. and Kirby, J., Quasiminimal structures and excellence, Bull. Lond. Math. Soc. 46(1) (2014), 155–163.Google Scholar
Buechler, S., Essential Stability Theory, Perspectives in Mathematical Logic (Springer, Berlin, 1996).10.1007/978-3-642-80177-8Google Scholar
Bays, M. and Zilber, B., Covers of multiplicative groups of algebraically closed fields of arbitrary characteristic, Bull. Lond. Math. Soc. 43(4) (2011), 689–702.Google Scholar
Fuchs, L., Infinite Abelian Groups, Vol. I, Pure and Applied Mathematics, Volume 36 (Academic Press, New York, 1970).Google Scholar
Gavrilovich, M., Model theory of the universal covering spaces of complex algebraic varieties. PhD thesis, Oxford University (2006).Google Scholar
Gavrilovich, M., A remark on transitivity of Galois action on the set of uniquely divisible abelian extensions in Ext1(E (ℚ), 𝛬), K-Theory 38(2) (2008), 135–152.Google Scholar
Hart, B., An exposition of OTOP, in Classification Theory (Chicago, IL, 1985), Lecture Notes in Math., Volume 1292, (Springer, Berlin, 1987), 107–126.Google Scholar
Jacquinot, O. and Ribet, K. A., Deficient points on extensions of abelian varieties by Gm, J. Number Theory 25(2) (1987), 133–151.Google Scholar
Lachlan, A. H., On the number of countable models of a countable superstable theory, in Logic, Methodology, Philosophy of Science 4 (North-Holland, Amsterdam, 1973), 45–56.Google Scholar
Lang, S., Diophantine geometry, in Number Theory III, Encyclopaedia of Mathematical Sciences, Volume 60, (Springer, Berlin, 1991).Google Scholar
Larsen, M., A mordell-weil theorem for abelian varieties over fields generated by torsion points, preprint 2005, arXiv:math/0503378v1 [math.NT].Google Scholar
Lascar, D., Les groupes 𝜔-stables de rang fini, Trans. Amer. Math. Soc. 292(2) (1985), 451–462.Google Scholar
Moosa, R. N., The model theory of compact complex spaces, in Logic Colloquium ’01, Lect. Notes Log., Volume 20, (Assoc. Symbol. Logic, Urbana, IL, 2005), 317–349.Google Scholar
Pillay, A. and Scanlon, T., Meromorphic groups, Trans. Amer. Math. Soc. 355(10) (2003), 3843–3859.Google Scholar
Ribet, K. A., Kummer theory on extensions of abelian varieties by tori, Duke Math. J. 46(4) (1979), 745–761.10.1215/S0012-7094-79-04638-6Google Scholar
Serre, J.-P., Quelques propriétés des variétés abéliennes en caractéristique p, Amer. J. Math. 80 (1958), 715–739.Google Scholar
Serre, J.-P., Œuvres. Collected papers. IV (Springer, Berlin, 2000), 1985–1998.Google Scholar
Shelah, S., Classification Theory and the Number of Nonisomorphic Models, Studies in Logic and the Foundations of Mathematics, Volume 92 (North-Holland Publishing Co., Amsterdam, 1990).Google Scholar
Zilber, B., Model theory, geometry and arithmetic of the universal cover of a semi-abelian variety, in Model Theory and Applications, Quad. Mat., Volume 11 (Aracne, Rome, 2002), 427–458.Google Scholar
Zilber, B., Covers of the multiplicative group of an algebraically closed field of characteristic zero, J. Lond. Math. Soc. (2) 74(1) (2006), 41–58.Google Scholar