Quantitative Reduction Theory and Unlikely Intersections. (16th July 2021)

• Record Type: Journal Article
• Authors: Daw, Christopher; Orr, Martin
• Is Part Of: International mathematics research notices. Volume 2022:Number 20(2022)
• Page Start: 16138
• Page End: 16195
• Publication Date: 2021-07-16
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnab173
• Languages: English
• ISSNs: 1073-7928

Abstract

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.