论文标题
$ j $ function的模糊
Blurrings of the $j$-function
论文作者
论文摘要
受指数函数模糊的想法的启发，我们定义了$ j $功能及其衍生物的模糊变体，其中模糊的是由$ \ rm {gl} _2 _2（\ mathbb {c}}）的子组的作用给出。对于密集的亚组（在复杂拓扑中），我们证明了存在的封闭定理，该定理指出，所有方程式都以相应的$ j $模糊为衍生物具有复杂的解决方案，除非有功能性超越原因，使它们不应该。对于没有衍生物的$ j $功能，我们证明了一个更强的定理，即$ j $的生存封闭性是由子组的作用模糊的，该子组以$ \ rm {gl} _2^+（\ mathbb {r}）的浓度致密，但不一定在$ \ rm \ rm {$ {$ {$ {$ {c.2（c）中。 我们还表明，对于一个适当选择的可计数代数封闭的子字段$ c \ subseteq \ mathbb {c} $，复杂字段增强了谓词，因为$ \ rm {gl} _2 _2（c）$模糊$ j $ function $ j $ function $ j $ j $ function在$ j $ function上，尤其是$ - $ - $ - $ -Stame。
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 $\rm{GL}_2(\mathbb{C})$. 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 $\rm{GL}_2^+(\mathbb{R})$, but not necessarily in $\rm{GL}_2(\mathbb{C})$. 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 $\rm{GL}_2(C)$ is model theoretically tame, in particular, $ω$-stable and quasiminimal.
