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NONDEFINABILITY RESULTS FOR ELLIPTIC AND MODULAR FUNCTIONS
- Part of
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 03 April 2024, pp. 1-20
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- Article
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Let
$\Omega $ be a complex lattice which does not have complex multiplication and 
$\wp =\wp _\Omega $ the Weierstrass 
$\wp $-function associated with it. Let 
$D\subseteq \mathbb {C}$ be a disc and 
$I\subseteq \mathbb {R}$ be a bounded closed interval such that 
$I\cap \Omega =\varnothing $. Let 
$f:D\rightarrow \mathbb {C}$ be a function definable in 
$(\overline {\mathbb {R}},\wp |_I)$. We show that if f is holomorphic on D then f is definable in 
$\overline {\mathbb {R}}$. The proof of this result is an adaptation of the proof of Bianconi for the 
$\mathbb {R}_{\exp }$ case. We also give a characterization of lattices with complex multiplication in terms of definability and a nondefinability result for the modular j-function using similar methods.
