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The Existential Closedness and Zilber-Pink Conjectures

Vahagn Aslanyan

Abstract

In this paper we survey the history of, and recent developments on, two major conjectures originating in Zilber's model-theoretic work on complex exponentiation -- Existential Closedness and Zilber-Pink. The main focus is on the modular versions of these conjectures and specifically on novel variants incorporating the derivatives of modular functions. The functional analogues of all the conjectures that we consider are theorems which are presented too. The paper also contains some new results and conjectures.

The Existential Closedness and Zilber-Pink Conjectures

Abstract

In this paper we survey the history of, and recent developments on, two major conjectures originating in Zilber's model-theoretic work on complex exponentiation -- Existential Closedness and Zilber-Pink. The main focus is on the modular versions of these conjectures and specifically on novel variants incorporating the derivatives of modular functions. The functional analogues of all the conjectures that we consider are theorems which are presented too. The paper also contains some new results and conjectures.
Paper Structure (16 sections, 13 theorems, 19 equations)

This paper contains 16 sections, 13 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Abbreviations
  3. Dedication
  4. The exponential setting
  5. Schanuel's conjecture and Exponential Closedness
  6. Conjecture on Intersections with Tori
  7. Functional/differential variants
  8. The modular setting
  9. Modular Schanuel conjecture and Modular Existential Closedness
  10. Modular Zilber-Pink
  11. Functional/differential variants
  12. Incorporating the derivatives of modular functions
  13. Modular Schanuel Conjecture with Derivatives
  14. Modular Existential Closedness with Derivatives
  15. Modular Zilber-Pink with Derivatives
  16. ...and 1 more sections

Key Result

Theorem 2.10

Let $(F;+,\cdot,D_1,\ldots,D_m)$ be a differential field with field of constants $C = \bigcap_{k=1}^m \ker D_k$. Let also $(x_i, y_i) \in F^2,~ i=1,\ldots,n,$ be such that $(\bar{x}, \bar{y})\in \mathop{\mathrm{Exp}}\nolimits(F)$. Assume $x_1,\ldots,x_n$ are $\mathop{\mathrm{\mathbb{Q}}}\nolimits$-l

Theorems & Definitions (58)

  • Conjecture 2.1: Schanuel's Conjecture, SC Lang-tr
  • Example 2.2: Aslanyan-Kirby-Mantova
  • Example 2.3
  • Conjecture 2.4: Exponential Closedness, EC Zilb-pseudoexpBays-Kirby-exp
  • Definition 2.5
  • Definition 2.6
  • Conjecture 2.7: Strong Exponential Closedness, SEC Zilb-pseudoexpBays-Kirby-exp
  • Remark 2.8
  • Conjecture 2.9: Conjecture on Intersections with Tori, CIT Zilb-exp-sum-publishedBom-Mas-ZanPila-ZP
  • Theorem 2.10: Ax-Schanuel Ax
  • ...and 48 more