Uniform Model Completeness for the Real Field with the Weierstrass $\wp$ Function

Abstract

In this work is we prove model completeness for the expansion of the real field by the Weierstrass $\wp$ function as a function of the variable $z$ and the parameter (or period) $\tau$. We need to existentially define the partial derivatives of the $\wp$ function with respect to the variable $z$ and the parameter $\tau$. In order to obtain this result we need to include in the structure function symbols for the unrestricted exponential function and restricted sine function, the Weierstrass $\zeta$ function and the quasimodular form $E_2$. 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 noneffectiveness of our proof and the difficulties to be overcome to obtain an effective model completeness result.