NONDEFINABILITY RESULTS FOR ELLIPTIC AND MODULAR FUNCTIONS

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 definablein (R, ℘|I ). We show that if f is holomorphic on D then f is definable in R. The proofof this result is an adaptation of the proof of Bianconi for the Rexp case. We also givea characterization of lattices with complex multiplication in terms of definability and a nondefinability result for the modular j-function using similar methods.<br/

Computability Theory

Computability is one of the fundamental notions of mathematics, trying to capture the effective content of mathematics. Starting from Gödel’s Incompleteness Theorem, it has now blossomed into a rich area with strong connections with other areas of mathematical logic as well as algebra and theoretical computer science

On local definability of holomorphic functions

Given a collection A of holomorphic functions, we consider how to describe all the holomorphic functions locally definable from A. The notion of local definability of holomorphic functions was introduced by Wilkie, who gave a complete description of all functions locally definable from A in the neighbourhood of a generic point. We prove that this description is no longer complete in the neighbourhood of non-generic points. More precisely, we produce three examples of holomorphic functions that suggest that at least three new operations need to be added to Wilkie's description in order to capture local definability in its entirety. The constructions illustrate the interaction between resolution of singularities and definability in the o-minimal setting

Local interdefinability of Weierstrass elliptic functions

We explain which Weierstrass elliptic functions are locally definable from other elliptic functions and exponentiation in the context of o-minimal structures. The proofs make use of the predimension method from model theory to exploit functional transcendence theorems in a systematic way