S-adic quadratic forms and Homogeneous Dynamics

Abstract: We present two new quantitative results about quadratic forms. Let $S = {\infty} \cup S_f$ be a finite set of places of Q. Consider the ring $Z_S$ of S-integers, and $Q_S = \prod{p \in S} Q_p$. The first is a solution to the problem of deciding if any given integral quadratic forms $Q_1$ and $Q_2$ are $Z_S$-equivalent. The proof is based on a reformulation of the problem in terms of the action of $O(Q_1, Q_S)$ on the space $X{d,S}$ of lattices of $Q_{S,d}$. A key tool are explicit mixing rates for the action of O(Q1, QS) on closed orbits in X{d,S}. As an application we obtain, for any S-integral orthogonal group, polynomial bounds on the S-norms of the elements of a finite generating set. These two results and the methods of proof are based on the work of H. Li and G. Margulis for $S = { \infty }$.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar