Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-24T22:53:32.518Z Has data issue: false hasContentIssue false
  • English
  • Français

Model Completeness for the Real Field with the Weierstrass ℘ Function

Part of: Other special functions Model theory Curves

Published online by Cambridge University Press:  21 May 2018

Ricardo Bianconi
Ricardo Bianconi*
Affiliation:
Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, Cidade Universitária, CEP 05508-090 São Paulo, Brazil ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove model completeness for the expansion of the real field by the Weierstrass ℘ function as a function of the variable z and the parameter (or period) τ. We need to existentially define the partial derivatives of the ℘ function with respect to the variable z and the parameter τ. To obtain this result, it is necessary to include in the structure function symbols for the unrestricted exponential function and restricted sine function, the Weierstrass ζ function and the quasi-modular form E2 (we conjecture that these functions are not existentially definable from the functions ℘ alone or even if we use the exponential and restricted sine functions). We prove some auxiliary model-completeness results with the same functions composed with appropriate change of variables. In the conclusion, we make some remarks about the non-effectiveness of our proof and the difficulties to be overcome to obtain an effective model-completeness result, and how to extend these results to appropriate expansion of the real field by automorphic forms.

Keywords

model completenessWeierstrass systemselliptic functions℘ functionWeierstrass ζ functionmodular formso-minimalitydefinable

MSC classification

Primary: 03C10: Quantifier elimination, model completeness and related topics
Secondary: 03C64: Model theory of ordered structures; o-minimality 03C98: Applications of model theory 14H52: Elliptic curves 33E05: Elliptic functions and integrals
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 3 , August 2018 , pp. 811 - 823
Copyright
Copyright © Edinburgh Mathematical Society 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

1.Abramowitz, M. and Stegun, I. A. (ed.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Ninth printing of 1965 edition (Dover Books on Mathematics, Dover Publications, New York, 1972).Google Scholar
2.Bianconi, R., Model completeness results for elliptic and abelian functions, Ann. Pure Appl. Logic 54(2) (1991), 121–136.CrossRefGoogle Scholar
3.Bianconi, R., Model complete expansions of the real field by modular functions and forms, SAJL 1(1) (2015), 321–335.Google Scholar
4.Denef, J. and van den Dries, L., p-adic and real subanalytic sets, Ann. of Math. (2) 128(1) (1988), 79–138.CrossRefGoogle Scholar
5.van den Dries, L., On the elementary theory of restricted elementary functions, J. Symbolic Logic 53(3) (1988), 796–808.CrossRefGoogle Scholar
6.van den Dries, L., Macintyre, A. and Marker, D., The elementary theory of restricted analytic fields with exponentiation, Ann. of Math. (2) 140(1) (1994), 183–205.CrossRefGoogle Scholar
7.Gabrielov, A. and Vorobjov, N., Complexity of computations with Pfaffian and Noetherian functions, in Normal forms, bifurcations and finiteness problems in differential equations, pp. 211–250 (Kluwer, 2004).CrossRefGoogle Scholar
8.Gao, Z., Towards the Andre–Oort conjecture for mixed Shimura varieties: the Ax–Lindemann theorem and lower bounds for Galois orbits of special points, J. Reine Angew. Math. 732 (2017), 85–146.CrossRefGoogle Scholar
9.Halphen, G.-H., Traité des fonctions Elliptiques et des leurs Applications, Première Partie: Théorie des Fonctions Elliptiques et des leurs Développements en Séries (Gauthier-Villars, Paris, 1886); https://archive.org/details/traitdesfonctio01halpgoog (accessed in June 2016).Google Scholar
10.Klingler, B., Ullmo, E. and Yafaev, A., The Hyperbolic Ax-Lindemann-Weierstrass conjecture. http://arxiv.org/abs/1307.3965.Google Scholar
11.Macintyre, A., The Elementary theory of elliptic functions I: the formalism and a special case, in O-minimal Structures, Lisbon 2003 (ed. Edmundo, M. J., Richardson, D. and Wilkie, A.). Proceedings of a Summer School by the European Research and Training Network, RAAG. http://www.maths.manchester.ac.uk/raag/preprints/0159.pdf (Real Algebraic and Analytic Geometry Preprint Server, University of Manchester; accessed in March 2014).Google Scholar
12.Macintyre, A., Some observations about the real and imaginary parts of complex Pfaffian functions, in Model theory with applications to algebra and analysis, Volume 1 (ed. Chatzidakis, Z., Macpherson, D., Pillay, A. and Wilkie, A.), pp. 215–223, London Mathematical Society Lecture Note Series, Volume 349 (Cambridge University Press, Cambridge, 2008).CrossRefGoogle Scholar
13.Macintyre, A. and Wilkie, A. J., On the decidability of the real exponential field, In Kreiseliana (ed. Odifreddi, P.), pp. 441–467 (A K Peters, Wellesley, MA, 1996).Google Scholar
14.Peterzil, Y. and Starchenko, S., Uniform definability of the Weierstrass ℘ functions and generalized tori of dimension one, Selecta Math. (N.S.) 10(4) (2004), 525–550.CrossRefGoogle Scholar
15.Peterzil, Y. and Starchenko, S., Definability of restricted theta functions and families of abelian varieties, Duke Math. J. 162(4) (2013), 731–765.CrossRefGoogle Scholar
16.Pila, J., O-minimality and the André–Oort conjecture for ℂn, Ann. of Math. (2) 173 (2011), 1779–1840.CrossRefGoogle Scholar
17.Pila, J. and Wilkie, A., The rational points of a definable set. Duke Math. J. 133 (2006), 591–616.CrossRefGoogle Scholar
18.Silverman, J. H., Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, Volume 151 (Springer-Verlag, New York, 1994).CrossRefGoogle Scholar
19.Whittaker, E. T. and Watson, G. N., A course on modern analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions. Reprint of the fourth (1927) edition. Cambridge Mathematical Library (Cambridge University Press, Cambridge, 1996).Google Scholar
20.Zagier, D., Elliptic modular forms and their applications, In The 1-2-3 of modular forms (ed. Ranestad, K.), Lectures at a Summer School in Nordfjordeid, Norway, pp. 1–103 (Springer-Verlag, Berlin, 2008).Google Scholar