Brauer equivalent number fields and the geometry of quaternionic Shimura varieties

TL;DR

This paper explores the properties of Brauer equivalent number fields and their implications for the geometry of quaternionic Shimura varieties, revealing new relationships between number theory and geometric structures.

Contribution

It establishes new number theoretic results about Brauer equivalent fields and applies them to construct incommensurable arithmetic locally symmetric spaces with identical geodesic surfaces.

Findings

Brauer equivalent fields share the same signature

Constructed incommensurable spaces with identical geodesic surfaces

Established links between number theory and geometric structures

Abstract

Two number fields are said to be Brauer equivalent if there is an isomorphism between their Brauer groups that commutes with restriction. In this paper we prove a variety of number theoretic results about Brauer equivalent number fields (e.g., they must have the same signature). These results are then applied to the geometry of certain arithmetic locally symmetric spaces. As an example, we construct incommensurable arithmetic locally symmetric spaces containing exactly the same set of proper immersed totally geodesic surfaces.