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Some new cases of Zilber-Pink in $Y(1)^3$

Christopher Daw, Martin Orr, Georgios Papas

TL;DR

The work proves the Zilber–Pink conjecture for curves in $Y(1)^3$ that meet a boundary modular curve, using a G-function framework to produce both $v$-adic and global relations among period-type data associated with families of elliptic schemes. It introduces a systematic construction of isogeny-based relations across multiple elliptic fibers, first for pairs then for triples, enabling uniform control over all places of good reduction. The authors derive height bounds for modular points, including a new unconditional finite-points result for points with few supersingular places, by combining local $v$-adic relations with global period conjectures and Pila–Zannier-type arguments. The work builds a robust interior analogue of Y(1) techniques, employing rigid-analytic neighbourhoods, Gauss–Manin connections, and isogeny estimates (e.g., Gaudron–Rémond) to achieve finiteness statements that mirror the boundary-degenerate cases in prior Andr%C3%A9–Oort–Pila–Zannier theories.

Abstract

We prove the Zilber-Pink conjecture for curves in $Y(1)^3$ that intersect a modular curve in the boundary. We also give an unconditional result for points having few places of supersingular reduction. Both results are proved using the G-function method for unlikely intersections.

Some new cases of Zilber-Pink in $Y(1)^3$

TL;DR

The work proves the Zilber–Pink conjecture for curves in that meet a boundary modular curve, using a G-function framework to produce both -adic and global relations among period-type data associated with families of elliptic schemes. It introduces a systematic construction of isogeny-based relations across multiple elliptic fibers, first for pairs then for triples, enabling uniform control over all places of good reduction. The authors derive height bounds for modular points, including a new unconditional finite-points result for points with few supersingular places, by combining local -adic relations with global period conjectures and Pila–Zannier-type arguments. The work builds a robust interior analogue of Y(1) techniques, employing rigid-analytic neighbourhoods, Gauss–Manin connections, and isogeny estimates (e.g., Gaudron–Rémond) to achieve finiteness statements that mirror the boundary-degenerate cases in prior Andr%C3%A9–Oort–Pila–Zannier theories.

Abstract

We prove the Zilber-Pink conjecture for curves in that intersect a modular curve in the boundary. We also give an unconditional result for points having few places of supersingular reduction. Both results are proved using the G-function method for unlikely intersections.

Paper Structure

This paper contains 31 sections, 18 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matrices
  4. Fields
  5. Elliptic curves
  6. Taylor series with respect to a local parameter
  7. G-functions
  8. G-functions associated with an abelian scheme
  9. Relations
  10. Simple neighbourhoods
  11. Relations for elliptic schemes
  12. Isogenous pairs
  13. Proof of Proposition \ref{['prop:rel']}
  14. Case 1
  15. Case 2
  16. ...and 16 more sections

Key Result

Theorem 3

Conjecture conj:ZPY(1) holds under the assumption that $\overline{C}\setminus C$ contains a modular point.

Theorems & Definitions (32)

  • Conjecture 1: Zilber--Pink
  • Conjecture 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Remark 7
  • Lemma 8
  • proof
  • Lemma 9
  • ...and 22 more