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Zilber-Pink in $Y(1)^n$: Beyond multiplicative degeneration

Georgios Papas

Abstract

We establish Large Galois orbits conjectures for points of unlikely intersections of curves in $Y(1)^n$, upon assumptions on the intersection of such curves with the boundary $X(1)^n\backslash Y(1)^n$, in the Zilber-Pink setting. As a corollary, building on work of Habegger-Pila and Daw-Orr, we obtain new cases of the Zilber-Pink conjecture for curves in $Y(1)^n$.

Zilber-Pink in $Y(1)^n$: Beyond multiplicative degeneration

Abstract

We establish Large Galois orbits conjectures for points of unlikely intersections of curves in , upon assumptions on the intersection of such curves with the boundary , in the Zilber-Pink setting. As a corollary, building on work of Habegger-Pila and Daw-Orr, we obtain new cases of the Zilber-Pink conjecture for curves in .
Paper Structure (18 sections, 18 theorems, 55 equations)

This paper contains 18 sections, 18 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Summary
  3. Notation
  4. Recollections on the G-functions method
  5. Families of tuples of elliptic curves
  6. Height bounds
  7. Reductions
  8. G-functions and relative periods
  9. Our setting
  10. Isogenies and archimedean relations
  11. Isogenies and periods
  12. The toy case: $n=3$
  13. The case $\mathbb{G}_m^2\times E$
  14. The $\mathbb{G}_m\times E\times E'$ case
  15. Archimedean relations at points with unlikely isogenies
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $C\subset Y(1)^n$ be an irreducible curve defined over $\bar{\mathbb{Q}}$ that is asymmetric and not contained in a proper special subvariety of $Y(1)^n$. Then the Zilber-Pink conjecture holds for $C$.

Theorems & Definitions (42)

  • Theorem 1.1: habeggerpila1, Theorem $1$
  • Theorem 1.2: daworr4, Theorem $1.3$
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Remark 2.2
  • Conjecture 2.3
  • Lemma 2.4
  • proof
  • ...and 32 more