Equations involving the modular $j$-function and its derivatives

TL;DR

The paper studies when algebraic equations involving the modular $j$-function and its derivatives have solutions that are Zariski-dense in the natural hypersurfaces defined by the equations. It develops a framework combining Rouché- and argument-principle methods with generic $ ext{SL}_2( olinebreak ext{Z})$-transforms and zero-estimates for polynomials in $(j,j',j'')$, to derive dense-sets results for a broad class of equations $F(z,j(z),j'(z),j''(z))=0$. A central achievement is showing that, for irreducible polynomials $F$ not depending only on $X$ and not divisible by the exceptional factors $Y_0$, $Y_0-1728$, or $Y_1$, the solution set is Zariski dense on the hypersurface $F=0$; more generally, the work yields concrete solvability criteria for equations involving periodic functions and extends these ideas to a wide range of periodic objects. The results provide a qualitative description of solution distribution and establish existential-closedness-with-derivatives in the one-variable setting, offering a flexible toolkit that applies beyond the $j$-function to other periodic or asymptotically periodic frameworks.

Abstract

We show that for any polynomial $F(X,Y_0,Y_1,Y_2) \in \mathbb{C}[X, Y_0, Y_1, Y_2]$, the equation $F(z,j(z),j'(z),j''(z))=0$ has a Zariski dense set of solutions in the hypersurface $F(X,Y_0,Y_1,Y_2)=0$, unless $F$ is in $\mathbb{C}[X]$ or it is divisible by $Y_0$, $Y_0-1728$, or $Y_1$. Our methods establish criteria for finding solutions to more general equations involving periodic functions. Furthermore, they produce a qualitative description of the distribution of these solutions.

Figures (3)

• Figure 1: The fundamental domains of the action by $\mathop{\mathrm{SL}}\nolimits_2(\mathop{\mathrm{\mathbb{Z}}}\nolimits)$, where $\mathbb{F}$ is highlighted by the striped background.
• Figure 2: The region $\Xi_r$ highlighted by the striped background.
• Figure 3: Visual representation of the action of $F_r$ on the boundary of a typical rectangle $\Xi_r$ for $F$ of the form $Az^2 + Bz + C{j(z)}^2 + Dj(z) + E$. The term $j^2$ has lowest order $(-2,0)$, so $F_r$ winds around $0$ twice while following the top side of the rectangle.

Theorems & Definitions (70)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Definition 1.5
• Proposition 1.6
• Example 1.7
• Definition 2.1
• Theorem 3.1: Rouché, see e.g. lang:complexanalysis
• Lemma 3.2