TY - JOUR

T1 - Pair arithmetical equivalence for quadratic fields

AU - Li, Wen Ching Winnie

AU - Rudnick, Zeev

N1 - Funding Information:
The research of Li is partially supported by Simons Foundation Grant # 355798. Z.R received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement no. 786758).
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature.

PY - 2021/10

Y1 - 2021/10

N2 - Given two nonisomorphic number fields K and M, and finite order Hecke characters χ of K and η of M respectively, we say that the pairs (χ, K) and (η, M) are arithmetically equivalent if the associated L-functions coincide: L(s,χ,K)=L(s,η,M).When the characters are trivial, this reduces to the question of fields with the same Dedekind zeta function, investigated by Gassmann in 1926, who found such fields of degree 180, and by Perlis (J Number Theory 9(3):342–360, 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.

AB - Given two nonisomorphic number fields K and M, and finite order Hecke characters χ of K and η of M respectively, we say that the pairs (χ, K) and (η, M) are arithmetically equivalent if the associated L-functions coincide: L(s,χ,K)=L(s,η,M).When the characters are trivial, this reduces to the question of fields with the same Dedekind zeta function, investigated by Gassmann in 1926, who found such fields of degree 180, and by Perlis (J Number Theory 9(3):342–360, 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.

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U2 - 10.1007/s00209-021-02706-w

DO - 10.1007/s00209-021-02706-w

M3 - Article

AN - SCOPUS:85101933360

SN - 0025-5874

VL - 299

SP - 797

EP - 826

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 1-2

ER -