Quantitative reduction theory and unlikely intersections

TL;DR

This paper develops quantitative reduction theory for arithmetic groups, providing polynomial bounds that enable applications to the Zilber--Pink conjecture in algebraic geometry, especially for abelian surfaces with special endomorphisms.

Contribution

It establishes polynomial bounds in reduction theory and applies these results to prove cases of the Zilber--Pink conjecture for abelian surfaces with quaternionic multiplication.

Findings

Polynomial bounds on reduced integral vectors

Polynomial bounds for fundamental sets of arithmetic subgroups

Proof of the Zilber--Pink conjecture under Galois orbits hypothesis

Abstract

We prove quantitative versions of Borel and Harish-Chandra's theorems on reduction theory for arithmetic groups. Firstly, we obtain polynomial bounds on the lengths of reduced integral vectors in any rational representation of a reductive group. Secondly, we obtain polynomial bounds in the construction of fundamental sets for arithmetic subgroups of reductive groups, as the latter vary in a real conjugacy class of subgroups of a fixed reductive group. Our results allow us to apply the Pila--Zannier strategy to the Zilber--Pink conjecture for the moduli space of principally polarised abelian surfaces. Building on our previous paper, we prove this conjecture under a Galois orbits hypothesis. Finally, we establish the Galois orbits hypothesis for points corresponding to abelian surfaces with quaternionic multiplication, under certain geometric conditions.