Abstract
AbstractWe 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.
Highlights
We prove quantitative versions of Borel and Harish-Chandra’s theorems on reduction theory for arithmetic groups
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
Reduction theory is concerned with finding small representatives for each orbit in actions of arithmetic groups, for example through constructing fundamental sets
Summary
Reduction theory is concerned with finding small representatives for each orbit in actions of arithmetic groups, for example through constructing fundamental sets. We note that the results on quantitative reduction theory in this paper will be an important tool for proving the Zilber–Pink conjecture for other Shimura varieties, which will be the subject of future work by the authors. We expect these results to have further applications, for example a uniform version of the second-named author’s bounds for polarisations and isogenies of abelian varieties [28] and bounds for the heights of generators of arithmetic groups by combining them with some of the techniques of homogeneous dynamics from [21]
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