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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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