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Preprint Number 1688
1688. Christopher Daw and Martin Orr
Quantitative reduction theory and unlikely intersections
Submission date: 13 November 2019
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 G(ℝ)--conjugacy class of subgroups of a fixed reductive group G.
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.
Mathematics Subject Classification: 11F06, 11G18
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