Locally Equivalent Correspondences

TL;DR

This paper establishes bijections between various algebraic and arithmetic objects associated with number fields having isomorphic rings of adeles, revealing deep structural similarities and invariants such as covolume and pro-congruence properties.

Contribution

It constructs explicit bijections between algebraic structures like Brauer groups, central simple algebras, and arithmetic lattices for number fields with isomorphic adele rings, extending previous finiteness results.

Findings

Bijections preserve covolume of lattices

Bijections commute with restriction maps in Brauer groups

Effective version of Prasad and Rapinchuk's finiteness result

Abstract

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 covolume and pro-congruence completion. We also make effective a finiteness result of Prasad and Rapinchuk.