Locally Equivalent Correspondences

Benjamin Linowitz, D. B. McReynolds, Nicholas Miller

Published 2015-05-18Version 1

Given a pair of number fields with isomorphic rings of adeles, we construct bijections between objects associated to the pair. For instance we construct an isomorphism of Brauer groups that commutes with restriction. We additionally construct bijections between central simple algebras, maximal orders, various Galois cohomology sets, and commensurability classes of arithmetic lattices in simple, inner algebraic groups. We show that under certain conditions, lattices corresponding to one another under our bijections have the same co-volume and pro-congruence completion. We also establish an effective version of a finiteness result of Prasad and Rapinchuk.