Joint IAS/Princeton University Number Theory Seminar

Number fields with the same Dedekind zeta function are said to be arithmetically equivalent. Such number fields necessarily have the same degree, discriminant, signature, Galois closure, and isomorphic unit groups, but may have different regulators, class groups, rings of adeles, and idele class groups. Motivated by a recent result of Prasad, I will discuss three stronger notions of arithmetic equivalence that force isomorphisms of some or all of these invariants without forcing an isomorphism of number fields, along with explicit examples and some open questions. 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).