Quantitative Reduction Theory and Unlikely Intersections

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