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Generic Solutions of Equations Involving the Modular $j$-function

Sebastian Eterović

TL;DR

This work advances the existential closedness program for the modular $j$-function by showing that, under a modular Schanuel-type conjecture (MSCD) and a modular Zilber–Pink conjecture (MZP), the strong EC problem reduces to the ordinary EC problem for broad and free varieties. It develops the machinery of convenient generators from aek-closureoperator, derives dimension-inequality controls over arbitrary finitely generated base fields, and uses Zilber–Pink to manage atypical intersections; these ingredients yield generic points on the graph of $j$ within $V$. The paper also provides unconditional versions when the ambient variety has no $C_j$-factors and explores a blurring of the $j$-function that yields approximate EC results; these ideas extend to derivatives of $j$ via the $J=(j,j',j'')$ framework and a two-sorted weak MZP approach. Collectively, the results offer a robust strategy for reducing strong EC questions for Shimura-type uniformization maps to EC-type problems, with potential impact on related areas such as Shimura varieties and dynamical systems arising from $j$-iterations.

Abstract

Assuming a modular version of Schanuel's conjecture and the modular Zilber-Pink conjecture, we show that the existence of generic solutions of certain families of equations involving the modular $j$ function can be reduced to the problem of finding a Zariski dense set of solutions. By imposing some conditions on the field of definition of the variety, we are also able to obtain unconditional versions of this result.

Generic Solutions of Equations Involving the Modular $j$-function

TL;DR

This work advances the existential closedness program for the modular -function by showing that, under a modular Schanuel-type conjecture (MSCD) and a modular Zilber–Pink conjecture (MZP), the strong EC problem reduces to the ordinary EC problem for broad and free varieties. It develops the machinery of convenient generators from aek-closureoperator, derives dimension-inequality controls over arbitrary finitely generated base fields, and uses Zilber–Pink to manage atypical intersections; these ingredients yield generic points on the graph of within . The paper also provides unconditional versions when the ambient variety has no -factors and explores a blurring of the -function that yields approximate EC results; these ideas extend to derivatives of via the framework and a two-sorted weak MZP approach. Collectively, the results offer a robust strategy for reducing strong EC questions for Shimura-type uniformization maps to EC-type problems, with potential impact on related areas such as Shimura varieties and dynamical systems arising from -iterations.

Abstract

Assuming a modular version of Schanuel's conjecture and the modular Zilber-Pink conjecture, we show that the existence of generic solutions of certain families of equations involving the modular function can be reduced to the problem of finding a Zariski dense set of solutions. By imposing some conditions on the field of definition of the variety, we are also able to obtain unconditional versions of this result.
Paper Structure (30 sections, 41 theorems, 123 equations)

This paper contains 30 sections, 41 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Summary of the proof of Theorem \ref{['thm:generic']}
  3. Strong EC for exp
  4. Structure of the paper
  5. Background
  6. Basic notation
  7. The j-function
  8. Modular polynomials
  9. Special points
  10. The modular Schanuel conjecture
  11. Existential Closedness and the Zilber--Pink Conjecture
  12. Broad and free varieties
  13. Existential Closedness
  14. Atypical intersections and Zilber--Pink
  15. Weak MZP
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $K\subseteq\mathbb{C}$ be a finitely generated field, and let $V\subseteq\mathbb{C}^{2n}$ be a broad and free variety defined over $K$ satisfying (EC). Then MSCD and MZP imply that $V$ has a point of the form $(z_1,\ldots,z_n,j(z_1),\ldots,j(z_n))$, with $(z_1,\ldots,z_n)\in\mathbb{H}^{n}$, whic

Theorems & Definitions (114)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: kirby-zilber
  • Definition
  • Definition
  • Remark 2.1
  • Conjecture 2.2: Modular Schanuel conjecture (MSC)
  • Conjecture 2.3: Modular Schanuel conjecture with derivatives (MSCD)
  • Remark 2.4
  • ...and 104 more