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A Common Generalisation of the André-Oort and André-Pink-Zannier Conjectures

Rodolphe Richard, Andrei Yafaev

Abstract

We introduce a ``hybrid'' conjecture which is a common generalisation of the André-Oort conjecture and the André-Pink-Zannier conjecture and we prove that it is a consequence of the Zilber-Pink conjecture. We also show that our hybrid conjecture implies the Zilber-Pink conjecture for hypersurfaces contained in weakly special subvarieties.

A Common Generalisation of the André-Oort and André-Pink-Zannier Conjectures

Abstract

We introduce a ``hybrid'' conjecture which is a common generalisation of the André-Oort conjecture and the André-Pink-Zannier conjecture and we prove that it is a consequence of the Zilber-Pink conjecture. We also show that our hybrid conjecture implies the Zilber-Pink conjecture for hypersurfaces contained in weakly special subvarieties.
Paper Structure (9 sections, 12 theorems, 52 equations)

This paper contains 9 sections, 12 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Shimura varieties and Weakly special subvarieties
  3. Various notions of Hecke orbits
  4. André-Pink-Zannier, André-Oort and Zilber-Pink conjectures
  5. Hybrid Hecke orbits and the Hybrid conjecture
  6. Intersection of a weakly special subvariety with special varieties
  7. Relation between the hybrid conjecture and the Zilber-Pink conjecture
  8. Relation to the work of Aslanyan and Daw
  9. Analogy with the Mordell-Lang conjecture

Key Result

Theorem 3.1

Any generalised Hecke orbit is a finite union of geometric Hecke orbits.

Theorems & Definitions (38)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 3.1
  • Definition 4: APZ1
  • Definition 3.2
  • Theorem 3.1: see APZ1
  • Conjecture 4.1: Generalised André-Pink-Zannier conjecture
  • Conjecture 4.2: André-Oort conjecture
  • Conjecture 4.3: Reformulation of André-Oort conjecture
  • ...and 28 more