Stronger arithmetic equivalence

TL;DR

This paper explores stronger notions of arithmetic equivalence among number fields, demonstrating their implications on invariants like class number and adele rings, and providing numerous explicit examples including a degree 96 case.

Contribution

It introduces and constructs infinitely many examples of solvable equivalence, expanding understanding of stronger arithmetic equivalences beyond prior group-theoretic examples.

Findings

Infinitely many examples of solvable equivalence are constructed.

Some examples include a degree 96 number field.

Results address questions posed by Scott, Guralnick, Weiss, and Prasad.

Abstract

Motivated by a recent result of Prasad, we consider three stronger notions of arithmetic equivalence: local integral equivalence, integral equivalence, and solvable equivalence. In addition to having the same Dedekind zeta function (the usual notion of arithmetic equivalence), number fields that are equivalent in any of these stronger senses must have the same class number, and solvable equivalence forces an isomorphism of adele rings. Until recently the only nontrivial example of integral and solvable equivalence arose from a group-theoretic construction of Scott that was exploited by Prasad. Here we provide infinitely many distinct examples of solvable equivalence, including a family that contains Scott's construction as well as an explicit example of degree 96. We also construct examples that address questions of Scott, and of Guralnick and Weiss, and shed some light on a question of Prasad.