Pair arithmetical equivalence for quadratic fields

DOI10.1007/S00209-021-02706-WarXiv2007.13147MaRDI QIDQ6345857

Wen-Ching Winnie Li, Zeév Rudnick

Publication date: 26 July 2020

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: ${\displaystyle }$L(s, chi, K) = L(s, eta, M) .${\displaystyle }$ 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.

Mathematics Subject Classification ID

Holomorphic modular forms of integral weight (11F11) Galois representations (11F80) Zeta functions and (L)-functions of number fields (11R42)

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