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

2025/07/07 by Bhatta, Bijay Raj
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.16827

Abstract

We prove the Zilber-Pink conjecture to the intersection of an irreducible Hodge generic algebraic subvariety V ⊂ A_g with special subvarieties of all simple PEL types other than 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.

Citations

• Unlikely intersections in Shimura varieties and beyond: a survey
• Abelian threefolds with imaginary multiplication
• The large Galois orbits conjecture under multiplicative degeneration
• Algebraic flows on Shimura varieties
• O-minimality and certain atypical intersections
• The Hyperbolic Ax-Lindemann-Weierstrass conjecture
• Galois orbits and equidistribution: towards the Andre-Oort conjecture
• Rational points in periodic analytic sets and the Manin-Mumford conjecture

Related

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• The Zilber-Pink conjecture for varieties not defined over \mathbb Q
• On the Geometric Zilber-Pink Theorem and the Lawrence-Venkatesh method
• On the Zilber-Pink conjecture for complex abelian varieties
• Modular Zilber-Pink for geometrically generic varieties