$j$ -功能模糊
Blurrings of the $j$-function
摘要：
受模糊指数函数的思想的启发，我们定义了模糊的$j$ -函数及其衍生物的变体，其中模糊是由$\rm{GL}_2(\mathbb{C})$的子群的作用给出的。对于一个稠密的子群（在复杂的拓扑），我们证明了一个潜在的封闭性定理，该定理指出，所有系统的方程在相应的模糊$j$与衍生物有复杂的解决方案，除了有一个功能超越的原因，为什么他们不应该。对于不含导数的$j$ -函数，我们证明了一个更强的定理，即$j$的势闭性被子群的作用所模糊，子群在$\rm{GL}_2^+(\mathbb{R})$中是稠密的，但不一定在$\rm{GL}_2(\mathbb{C})$中是稠密的.我们还表明，对于一个适当选择的可数代数闭子域$C \subseteq \mathbb{C}$，增加了一个谓词的$j$ -功能模糊的$\rm{GL}_2(C)$的复杂的领域是模型理论上驯服，特别是，$ω$ -稳定和拟最小。
英文摘要：
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.
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