Stronger arithmetic equivalence

Abstract: Number fields K1 and K2 with the same Dedekind zeta function are said to be arithmetically equivalent. Such number fields necessarily have the same degree, signature, unit group, discriminant, and Galois closure, and the distributions of their Frobenius elements are compatible in a strong sense: for every unramified prime p the base change of the Q-algebras K1 and K2 to Qp are isomorphic. This need not hold at ramified primes, so the adele rings of K1 and K2 need not be isomorphic, and global invariants such as the regulator and class number may differ.

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, and present examples that address questions of Scott and of Guralnick and Weiss, and shed some light on a question of Prasad. These results also have applications to the construction of curves with the same L-function (due to Prasad), isospectral Riemannian manifolds (due to Sunada), and isospectral graphs (due to Halbeisen and Hungerbuhler).

Preprint: arxiv.org/abs/2104.01956

number theory

Audience: researchers in the topic

( paper | slides )

Around Frobenius distributions and related topics II