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 (GL)2 (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 (GL)2+ (ℝ;)), but not necessarily in (GL)2 (C))). We also show that for a suitably chosen countable algebraically closed subfield C (C)), the complex field augmented with a predicate for the blurring of the j-function by (GL)2 (C) is model theoretically tame, in particular, ω-stable and quasiminimal.
Original language | English |
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Pages (from-to) | 461–475 |
Number of pages | 15 |
Journal | The Quarterly Journal of Mathematics |
Volume | 73 |
Issue number | 2 |
Early online date | 14 Aug 2021 |
DOIs | |
Publication status | Published - Jun 2022 |