Effective Estimates on Integral Quadratic Forms: Masser’s Conjecture, Generators of Orthogonal Groups, and Bounds in Reduction Theory

Abstract

In this paper we prove a conjecture of David Masser on small height integral equivalence between integral quadratic forms. Using our resolution of Masser’s conjecture we show that integral orthogonal groups are generated by small elements which is essentially an effective version of Siegel’s theorem on the finite generation of these groups. We also obtain new estimates on reduction theory and representation theory of integral quadratic forms. Our line of attack is to make and exploit the connections between certain problems about quadratic forms and group actions, whence we may study the problem in the well-developed theory of homogeneous dynamics, arithmetic groups, and the spectral theory of automorphic forms.