Blurrings Of The J-Function

Abstract

Abstract Inspired by the idea of blurring the exponential function, we define blurred variants of the j-function and its derivatives, where blurring is given by the action of a subgroup of $\mathrm{GL}_2({\mathbb{C}})$. For a dense subgroup (in the complex topology) we prove an Existential Closedness theorem which states that all systems of equations in terms of the corresponding blurred j with derivatives have complex solutions, except where there is a functional transcendence reason why they should not. For the j-function without derivatives we prove a stronger theorem, namely, Existential Closedness for j blurred by the action of a subgroup which is dense in $\mathrm{GL}_2^+(\mathbb{R})$, but not necessarily in $\mathrm{GL}_2({\mathbb{C}})$. We also show that for a suitably chosen countable algebraically closed subfield $C \subseteq {\mathbb{C}}$, the complex field augmented with a predicate for the blurring of the j-function by $\mathrm{GL}_2(C)$ is model theoretically tame, in particular, ω-stable and quasiminimal.