S-integral quadratic forms and homogeneous dynamics

2022/02/21 by Calderón, Irving
#11E12 #11H55 #37A17 (Primary) 11E08 #37A25 (Secondary) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2202.10257

Abstract

Let S = \ ∞ \ ∪ S_f be a finite set of places of ℚ. Using homogeneous dynamics, we establish two new quantitative and explicit results about integral quadratic forms in three or more variables: The first is a criterion of S-integral equivalence. The second determines a finite generating set of any S-integral orthogonal group. Both theorems--which extend results of H. Li and G. Margulis for S = \ ∞\--are given by polynomial bounds on the size of the coefficients of the quadratic forms.

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