Title: Strong Arithmetic Equivalence

Abstract: Number fields with the same Dedekind zeta function are said to be arithmetically equivalent. Such number fields necessarily have the same degree, discriminant, signature, and Galois closure, and they also have isomorphic unit groups, but may have different regulators, class groups, rings of adeles, and idele class groups, and ramification indices may differ. Motivated by a recent result of Prasad, I will discuss three stronger notions of arithmetic equivalence that force isomorphism of some or all of these invariants without forcing an
isomorphism of number fields, along with explicit examples. These results also have applications to the construction of curves with isomorphic Jacobians (due to Prasad), isospectral Riemannian manifolds (due to Sunada), and isospectral graphs (due to Halbeisen and Hungerbuhler).