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Model Completeness for the Real Field with the Weierstrass ℘ Function

Part of: Other special functions Curves Model theory

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])
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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 

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