NONDEFINABILITY RESULTS FOR ELLIPTIC AND MODULAR FUNCTIONS

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Abstract

Let Ω be a complex lattice which does not have complex multiplication and ℘ = ℘Ω the Weierstrass ℘-function associated to it. Let D ⊆ C be a disc and I ⊆ R be a bounded closed interval such that I ∩ Ω = ∅. Let f : D → C be a function definable
in (R, ℘|I ). We show that if f is holomorphic on D then f is definable in R. The proof
of this result is an adaptation of the proof of Bianconi for the Rexp 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.
Original languageEnglish
Pages (from-to)1-18
JournalThe Journal of Symbolic Logic
Early online date3 Apr 2024
DOIs
Publication statusE-pub ahead of print - 3 Apr 2024

Keywords

  • Model theory
  • Weierstrass ℘-function
  • modular j-function
  • nondefinability
  • Ax-Schanuel theorem

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