Modular Existential Closedness

We prove the Modular Existential Closedness conjecture for the modular $j$-function extended from the upper half-plane to $\mathbb{C} \setminus \mathbb{R}$ by Schwarz reflection. Specifically, we show that any system of equations involving the field operations and $j$ has solutions in the complex numbers unless it is overdetermined. Such a system is overdetermined if, roughly, there are more independent equations than variables, possibly after applying the transformation rules of the $j$-function under the linear fractional action of $\rm{PGL}_2(\mathbb{Q})$ on $\mathbb{C} \setminus \mathbb{R}$. In general, we do not expect overdetermined systems of equations to have solutions due to the algebraic and transcendental properties of the $j$-function captured by the modular analogue of Schanuel's conjecture. In geometric terms, Modular Existential Closedness states that given a broad and free irreducible algebraic variety $V \subseteq \mathbb{C}^n \times \mathbb{C}^n$ -- where broadness and freeness are geometric conditions guaranteeing that the system is not overdetermined -- there is a point of the form $(τ_1, \ldots, τ_n, j(τ_1), \ldots, j(τ_n)) \in V$ with $τ_i \in \mathbb{C} \setminus \mathbb{R}$. The proof uses Ratner's theorems on the closure and equidistribution of the orbits of unipotent flows in powers of $\rm{PGL}_2(\mathbb{R})$ to describe the asymptotic behaviour of the orbits of certain analytic maps. This is then applied to approximate a relatively simple system of equations using the original system and the action of $\rm{PGL}_2(\mathbb{Z})$. The construction of the limiting system makes use of the Ax--Schanuel theorem and guarantees an isolated solution. Rouché's theorem then establishes the existence of solutions for the original system of equations.