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Abelian fields and the Brumer-Stark conjecture. (English) Zbl 0552.12007
Let \(K/k\) be a finite abelian extension of number fields. For a character \(\chi\) of \(\text{Gal}(K/k)\), let \(L_ S(s,\chi)\) denote the Artin \(L\)-function with Euler factors for primes in a set \(S\) removed (\(S\) contains at least the infinite primes and the primes which ramify in \(K\)). The author studies the following Brumer-Stark conjecture: For each ideal \({\mathfrak a}\) of \(K\) there exists an element \(\varepsilon({\mathfrak a})\) in \(K\) such that (1) \(\varepsilon({\mathfrak a})\) has absolute value 1 at all infinite primes of \(K\), (2) the principal ideal generated by \(\varepsilon({\mathfrak a})\) equals \({\mathfrak a}^{w\theta}\) where \(w\) is the number of roots of unity in \(K\) and \(\theta =\theta_ S(K/k)\) is the Stickelberger element, (3) \(K(\varepsilon ({\mathfrak a})^{1/w})\) is an abelian extension of \(k\).
He proves this conjecture when \(K\) is abelian over \(\mathbb Q\) (\(K\) is contained in the \(f\)-th cyclotomic field, say), \(k\) is any totally real subfield of \(K\) and \(S\) is the set of primes of \(k\) which are either infinite or divisors of \(f\). For each \({\mathfrak a}\) relatively prime to \(2f\), he gives \(\varepsilon({\mathfrak a})\) explicitly as a product of powers of Gauss sums and shows that \({\mathfrak a}\to \varepsilon({\mathfrak a})\) is a Hecke character with defining ideal \((f)^ 2\) or \((2f)^ 2\) according as\( f\) is even or odd. His method also gives a simple verification of the integrality results of P. Deligne and K. Ribet [Invent. Math. 59, 227–286 (1980; Zbl 0434.12009)] for the field extensions under consideration.

MSC:
11R42 Zeta functions and \(L\)-functions of number fields
11R18 Cyclotomic extensions
11L10 Jacobsthal and Brewer sums; other complete character sums
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References:
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