Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-05T15:26:20.712Z Has data issue: false hasContentIssue false
  • English
  • Français

LOCAL INTERDEFINABILITY OF WEIERSTRASS ELLIPTIC FUNCTIONS

Part of: Model theory Number theory: Connections with logic

Published online by Cambridge University Press:  24 November 2014

Gareth Jones ,
Jonathan Kirby  and
Tamara Servi
Gareth Jones
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK ([email protected])
Jonathan Kirby
Affiliation:
School of Mathematics, University of East Anglia, Norwich Research Park, Norwich NR4 7TJ, UK ([email protected])
Tamara Servi
Affiliation:
Centro de Matemática e Applicações Fundamentais, Av. Prof. Gama Pinto, 2 1649-003, Lisboa, Portugal ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We explain which Weierstrass ${\wp}$-functions are locally definable from other ${\wp}$-functions and exponentiation in the context of o-minimal structures. The proofs make use of the predimension method from model theory to exploit functional transcendence theorems in a systematic way.

Keywords

o-minimalitylocal definabilityWeierstrass ℘-functionpredimension

MSC classification

Primary: 03C64: Model theory of ordered structures; o-minimality
Secondary: 11U09: Model theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 15 , Issue 4 , October 2016 , pp. 673 - 691
Copyright
© Cambridge University Press 2014 

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

Ax, J., On Schanuel’s conjectures, Ann. of Math. (2) 93 (1971), 252–268.CrossRefGoogle Scholar
Ax, J., Some topics in differential algebraic geometry. I. Analytic subgroups of algebraic groups, Am. J. Math. 94 (1972), 1195–1204.Google Scholar
Bertolin, C., Périodes de 1-motifs et transcendance, J. Number Theory 97(2) (2002), 204–221.CrossRefGoogle Scholar
Bianconi, R., Nondefinability results for expansions of the field of real numbers by the exponential function and by the restricted sine function, J. Symb. Logic 62(4) (1997), 1173–1178.CrossRefGoogle Scholar
van den Dries, L. and Miller, C., Extending Tamm’s theorem, Ann. Inst. Fourier (Grenoble) 44(5) (1994), 1367–1395.Google Scholar
van den Dries, L. and Miller, C., On the real exponential field with restricted analytic functions, Israel J. Math. 85(1–3) (1994), 19–56.Google Scholar
Eisenbud, D., Commutative Algebra, Graduate Texts in Mathematics, Volume 150 (Springer-Verlag, New York, 1995).CrossRefGoogle Scholar
Gabrielov, A., Complements of subanalytic sets and existential formulas for analytic functions, Invent. Math. 125(1) (1996), 1–12.Google Scholar
Hrushovski, E., A new strongly minimal set, Ann. Pure Appl. Log. 62(2) (1993), 147–166.Google Scholar
Jones, G. O. and Thomas, M. E. M., The density of algebraic points on certain Pfaffian surfaces, Q. J. Math. 63(3) (2012), 637–651.CrossRefGoogle Scholar
Jones, G. O. and Wilkie, A. J., Locally polynomially bounded structures, Bull. Lond. Math. Soc. 40(2) (2008), 239–248.Google Scholar
Kirby, J., Exponential and Weierstrass equations, Presented at the Workshop: An Introduction to Recent Applications of Model Theory, April 2005.Google Scholar
Kirby, J., The theory of the exponential differential equations of semiabelian varieties, Selecta Math. (N.S.) 15(3) (2009), 445–486.CrossRefGoogle Scholar
Kirby, J., Exponential algebraicity in exponential fields, Bull. London Math. Soc. 42(5) (2010), 879–890.CrossRefGoogle Scholar
Lang, S., Algebra, 3rd edn, Graduate Texts in Mathematics, Volume 211 (Springer, New York, 2002).Google Scholar
Macpherson, D., Notes on o-minimality and variations, in Model Theory, Algebra, and Geometry, Mathematical Sciences Research Institute Publications, Volume 39, pp. 97–130 (Cambridge University Press, Cambridge, 2000).Google Scholar
Macintyre, A., Some observations about the real and imaginary parts of complex Pfaffian functions, in Model Theory with Applications to Algebra and Analysis, Vol. 1, London Mathematical Society Lecture Note Series, Volume 349, pp. 215–223 (Cambridge University Press, Cambridge, 2008).CrossRefGoogle Scholar
Masser, D., Heights, transcendence, and linear independence on commutative group varieties, in Diophantine Approximation (Cetraro, 2000), Lecture Notes in Mathematics, Volume 1819, pp. 1–51 (Springer, Berlin, 2003).Google Scholar
Mumford, D., Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, Volume 5 (Tata Institute of Fundamental Research, Bombay, 1970).Google Scholar
Masser, D. W. and Wüstholz, G., Zero estimates on group varieties, II, Invent. Math. 80(2) (1985), 233–267.Google Scholar
Pila, J., O-minimality and the André-Oort conjecture for ℂ n , Ann. Math. (2) 173(3) (2011), 1779–1840.Google Scholar
Pila, J. and Wilkie, A. J., The rational points of a definable set, Duke Math. J. 133(3) (2006), 591–616.Google Scholar
Silverman, J. H., The arithmetic of elliptic curves, 2nd edn, Graduate Texts in Mathematics, Volume 106 (Springer, Dordrecht, 2009).Google Scholar
Tamm., M., Subanalytic sets in the calculus of variation, Acta Math. 146(3–4) (1981), 167–199.Google Scholar
Wilkie, A. J., Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Am. Math. Soc. 9(4) (1996), 1051–1094.CrossRefGoogle Scholar
Wilkie, A. J., Liouville functions, in Logic Colloquium 2000, Lecture Notes in Logic, Volume 19, pp. 383–391 (Association for Symbolic Logic, Urbana, IL, 2005).Google Scholar
Wilkie, A. J., Covering definable open sets by open cells, in Proceedings of the RAAG Summer School Lisbon 2003 o-Minimal Structures, Lecture Notes in Real Algebraic and Analytic Geometry, pp. 32–36 (Cuvillier Verlag, Göttingen, 2005).Google Scholar
Wilkie, A. J., Some local definability theory for holomorphic functions, in Model Theory with Applications to Algebra and Analysis, Vol. 1, LMS Lecture Note Series, Volume 349, pp. 197–213 (Cambridge, 2008).Google Scholar
Zilber, B., Exponential sums equations and the Schanuel conjecture, J. London Math. Soc. (2) 65(1) (2002), 27–44.Google Scholar
Zilber, B., A theory of a generic function with derivations, in Logic and Algebra, Contemporary Mathematics, Volume 302, pp. 85–99 (American Mathematical Society, Providence, RI, 2002).Google Scholar