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Parameter Height bounds for the Zilber Pink conjecture for PEL types III and IV

Bijay Raj Bhatta

TL;DR

This work establishes a conditional Zilber–Pink statement for the intersection of an irreducible Hodge-generic subvariety of $\\mathcal{A}_g$ with simple PEL-type III/IV special subvarieties, assuming the Large Galois Orbits conjecture. It introduces a parameter-height bound for the arithmetic inputs of the Pila–Zannier strategy in Albert types III and IV, and develops a unitary-basis framework to control discriminants and covolumes. Extending prior types I/II results, the paper obtains finiteness and non-density conclusions for curves and more general subvarieties, linking Shimura data, o-minimality, and height bounds. The methods combine a refined classification of conjugacy classes, a definable-sets counting approach, and inductive height-control techniques that may inform unconditional progress via LGO in broader PEL contexts.

Abstract

We prove the Zilber-Pink conjecture to the intersection of an irreducible Hodge generic algebraic subvariety $ V \subset \mathcal{A}_g$ with special subvarieties of all simple PEL types other than $\mathbb{Z}$, under the assumption of the Large Galois Orbits conjecture. In particular, we establish parameter height bounds for the arithmetic ingredients of the Pila-Zannier strategy in the case of Albert types III and IV. This paper is a sequel to Daw and Orr's paper "Lattices with skew-Hermitian forms over division algebras and unlikely intersections" 2023.

Parameter Height bounds for the Zilber Pink conjecture for PEL types III and IV

TL;DR

This work establishes a conditional Zilber–Pink statement for the intersection of an irreducible Hodge-generic subvariety of with simple PEL-type III/IV special subvarieties, assuming the Large Galois Orbits conjecture. It introduces a parameter-height bound for the arithmetic inputs of the Pila–Zannier strategy in Albert types III and IV, and develops a unitary-basis framework to control discriminants and covolumes. Extending prior types I/II results, the paper obtains finiteness and non-density conclusions for curves and more general subvarieties, linking Shimura data, o-minimality, and height bounds. The methods combine a refined classification of conjugacy classes, a definable-sets counting approach, and inductive height-control techniques that may inform unconditional progress via LGO in broader PEL contexts.

Abstract

We prove the Zilber-Pink conjecture to the intersection of an irreducible Hodge generic algebraic subvariety with special subvarieties of all simple PEL types other than , under the assumption of the Large Galois Orbits conjecture. In particular, we establish parameter height bounds for the arithmetic ingredients of the Pila-Zannier strategy in the case of Albert types III and IV. This paper is a sequel to Daw and Orr's paper "Lattices with skew-Hermitian forms over division algebras and unlikely intersections" 2023.

Paper Structure

This paper contains 16 sections, 39 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Structure of the Paper
  3. Acknowledgements
  4. Division algebras
  5. Preliminary Lemmas
  6. Unitary bases and signature
  7. Orders and discriminants
  8. Anti-symmetric elements in division algebras of types III and IV
  9. Cases of Zilber-Pink
  10. Shimura data
  11. Conjugacy classes of simple PEL-III and IV type
  12. Proof of Theorem \ref{['zp1']}
  13. Proof of Theorem \ref{['ZP-end']}
  14. Proof of Theorem \ref{['cor2']}
  15. Proof of Theorem \ref{['height-bound']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $g \ge 3$. Let $\Sigma$ be the set of points $s \in \mathcal{A}_g(\mathbb{C})$ such that the associated abelian variety $A_s$ has endomorphism algebra which is a division algebra of Albert type III or IV. Let $C$ be an irreducible Hodge generic algebraic curve in $\mathcal{A}_g$. If $C$ satisfie

Theorems & Definitions (73)

  • Theorem 1.1
  • Conjecture 1
  • Theorem 1.2
  • Conjecture 2
  • Theorem 1.3
  • Corollary 1
  • proof
  • Theorem 1.4
  • proof
  • Theorem 1.5
  • ...and 63 more