Pair arithmetical equivalence for quadratic fields

Abstract: Given two distinct number fields $K$ and $M$, and two 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 Gassmann in 1926, who found such fields of degree 180, and by Perlis in 1977 and others, who showed that there are no nonisomorphic fields of degree less than 7.

In this talk we discuss arithmetically equivalent pairs where the fields are quadratic. They give rise to dihedral automorphic forms induced from characters of different quadratic fields. We characterize when a given pair is arithmetically equivalent to another pair, explicitly construct such pairs for infinitely many quadratic extensions with odd class number, and classify such characters of order 2.

This is a joint work with Zeev Rudnick.

algebraic geometrynumber theory

Audience: researchers in the topic

( paper )

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MIT number theory seminar

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