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Density theorems for reciprocity equivalences. (English) Zbl 0923.11066

Let the pair \((t,T)\) be a reciprocity equivalence between algebraic number fields \(K\) and \(L\). So \(T:\Omega_K\to\Omega_L\) is a bijection between the sets of primes of \(K\) and \(L\), and \(t:\dot K/\dot K^2\to \dot L/\dot L^2\) is a group isomorphism such that Hilbert symbols are preserved. The equivalence is said to be tame at a finite prime \(P\) of \(K\) if the parity of \(P\)-orders of elements is preserved under the equivalence; otherwise, the equivalence is said to be wild at \(P\). The set of all \(P\) at which \((t,T)\) is wild is called the wild set of the equivalence.
With regard to the wild set, it is proved here that 1) the Dirichlet density of the wild set is zero, and 2) between two reciprocity equivalent number fields, there exists a reciprocity equivalence with an infinite wild set. Additionally, it is shown that between any pair of reciprocity equivalent fields, the mapping \(T\) determines the mapping \(t\), and the mapping \(T\) is determined (except possibly at the complex primes) by its action on any set of primes of positive density. The proofs build upon the results of a fundamental paper of R. Perlis, K. Szymiczek, P. E. Conner and R. Litherland [Contemp. Math. 155, 365-387 (1994; Zbl 0807.11024)].

MSC:

11E81 Algebraic theory of quadratic forms; Witt groups and rings
11E12 Quadratic forms over global rings and fields
11R45 Density theorems

Citations:

Zbl 0807.11024
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