Pair arithmetical equivalence for quadratic fields

TL;DR

This paper constructs infinitely many pairs of quadratic fields with arithmetically equivalent L-functions, extending classical results and providing new examples of dihedral automorphic forms induced from different quadratic fields.

Contribution

It introduces a method to generate infinitely many pairs of quadratic fields with identical L-functions, expanding the understanding of arithmetical equivalence beyond trivial characters.

Findings

Constructed infinitely many pairs of quadratic fields with arithmetically equivalent L-functions.

Classified order 2 characters of quadratic fields with odd class number in these examples.

Extended classical results on fields with the same Dedekind zeta function.

Abstract

Given two distinct number fields $K$ and $M$, and finite order Hecke characters $\chi$ of $K$ and $\eta$ of $M$ respectively, we say that the pairs $(\chi, K)$ and $(\eta, M)$ are arithmetically equivalent if the associated L-functions coincide: $$L(s, \chi, K) = L(s, \eta, M) .$$ When the characters are trivial, this reduces to the question of fields with the same Dedekind zeta function, investigated by Gassman in 1926, who found such fields of degree 180, and by Perlis (1977) and others, who showed that there are no nonisomorphic fields of degree less than $7$. We construct infinitely many such pairs where the fields are quadratic. This gives dihedral automorphic forms induced from characters of different quadratic fields. We also give a classification of such characters of order 2 for the quadratic fields of our examples, all with odd class number.