A refined notion of arithmetically equivalent number fields, and curves with isomorphic Jacobians

TL;DR

This paper constructs examples of non-isomorphic number fields and algebraic curves that have isomorphic idele class groups and Jacobian varieties respectively, using group-theoretic methods involving non-conjugate subgroups.

Contribution

It introduces a refined notion of arithmetically equivalent fields and curves with isomorphic Jacobians, based on specific group-theoretic constructions.

Findings

Number fields with isomorphic idele class groups but non-isomorphic.

Curves with isomorphic Jacobians but non-isomorphic.

Utilizes non-conjugate subgroups with isomorphic permutation modules.

Abstract

We construct examples of number fields which are not isomorphic but for which their idele class groups are isomorphic. We also construct examples of projective algebraic curves which are not isomorphic but for which their Jacobian varieties are isomorphic. Both are constructed using an example in group theory provided by Leonard Scott of a finite group $G$ and subgroups $H_1$ and $H_2$ which are not conjugate in $G$ but for which the $G$-module ${\mathbb Z}[G/H_1]$ is isomorphic to ${\mathbb Z}[G/H_2]$.