Counting rational matrices of a fixed irreducible characteristic polynomial.

摘要

The number of rational matrices of a given rational characteristic polynomial and of bounded height is finite. We establish the asymptotic growth for this number as height increases, assuming the characteristic polynomial is irreducible. We thus prove a new case of Manin's Conjecture.;The method of proof is inspired by the counting integral points theorem of Eskin-Mozes-Shah. However, we do not make use of unipotent flows nor Ratner's Theorems. Instead, we rely on the measure rigidity theorem for maximal torus orbits of Einsiedler-Katok-Lindenstrauss.;The counting estimate is a consequence of a new adelic mean ergodic theorem. Given a Q -anisotropic maximal torus H in PGLn , the group of its adelic points has a compact orbit through PGLn( Q ) in XA = PGLn( Q )PGLn( A ). We prove that the average of its translates over the height-ball equidistributes over XA . To rule out intermediary measures, we will make use of the shape of the height-ball and tranverse trajectories to the H-orbits.