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Model Completeness for the Real Field with the Weierstrass ℘ Function

  • Ricardo Bianconi (a1)
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.

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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).
2.Bianconi, R., Model completeness results for elliptic and abelian functions, Ann. Pure Appl. Logic 54(2) (1991), 121136.
3.Bianconi, R., Model complete expansions of the real field by modular functions and forms, SAJL 1(1) (2015), 321335.
4.Denef, J. and van den Dries, L., p-adic and real subanalytic sets, Ann. of Math. (2) 128(1) (1988), 79138.
5.van den Dries, L., On the elementary theory of restricted elementary functions, J. Symbolic Logic 53(3) (1988), 796808.
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), 183205.
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. 211250 (Kluwer, 2004).
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), 85146.
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).
10.Klingler, B., Ullmo, E. and Yafaev, A., The Hyperbolic Ax-Lindemann-Weierstrass conjecture. http://arxiv.org/abs/1307.3965.
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).
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. 215223, London Mathematical Society Lecture Note Series, Volume 349 (Cambridge University Press, Cambridge, 2008).
13.Macintyre, A. and Wilkie, A. J., On the decidability of the real exponential field, In Kreiseliana (ed. Odifreddi, P.), pp. 441467 (A K Peters, Wellesley, MA, 1996).
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), 525550.
15.Peterzil, Y. and Starchenko, S., Definability of restricted theta functions and families of abelian varieties, Duke Math. J. 162(4) (2013), 731765.
16.Pila, J., O-minimality and the André–Oort conjecture for ℂn, Ann. of Math. (2) 173 (2011), 17791840.
17.Pila, J. and Wilkie, A., The rational points of a definable set. Duke Math. J. 133 (2006), 591616.
18.Silverman, J. H., Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, Volume 151 (Springer-Verlag, New York, 1994).
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).
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. 1103 (Springer-Verlag, Berlin, 2008).
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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
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