zbMATH — the first resource for mathematics

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.

11R42 Zeta functions and \(L\)-functions of number fields
11R18 Cyclotomic extensions
11L10 Jacobsthal and Brewer sums; other complete character sums
Full Text: Numdam EuDML
[1] P. Deligne and K. Ribet , Values of abelian L-functions at negative integers over totally real fields . Inventiones Mathematicae 59 (1980) 227-286. · Zbl 0434.12009 · doi:10.1007/BF01453237 · eudml:142740
[2] B.H. Gross , P-adic L-series at s = 0 , manuscript for J. Fac. Sci., U. Tokyo, Sect. IA 28 (1981), No. 3, 979-994. · Zbl 0507.12010
[3] A. Hurwitz , Einige Eigenschaften Dirichlet’schen Funktionen . Zeit. fur Math. Phys. 27 (1882) 86-101(Math Werke I, 72-88). · JFM 14.0371.01
[4] D. Kubert and S. Lichtenbaum , Jacobi-sum Hecke characters and Gauss sum identities . Comp Math. 48 (1983) Fasc. 1, 55-87. · Zbl 0513.12010 · numdam:CM_1983__48_1_55_0 · eudml:89587
[5] S. Lang , Cyclotomic Fields . Springer-Verlag, New York (1978). · Zbl 0395.12005
[6] D. Rideout , A generalization of Stickelbergers’ Theorem . Ph.D. Thesis, McGill, Montreal (1970).
[7] J.W. Sands , The Conjecture of Gross and Stark for Special Values of Abelian L-series over Totally Real Fields . Ph.D. Thesis, U.C.S.D., San Diego (1982).
[8] H.M. Stark , L-functions at s =1. IV. First derivatives at s = 0 , Advances in Math. 35 (1980) 197-235. · Zbl 0475.12018 · doi:10.1016/0001-8708(80)90049-3
[9] H.M. Stark , Values of Zeta and L-functions , to appear in proceedings of conference to honor Dedekind’s 150th birthday. · Zbl 0509.12012
[10] J. Tate , Brumer-Stark-Stickelberger, Seminaire de Theorie des Nombres Annee 1980-81 , expose no. 24. · Zbl 0504.12005 · eudml:182099
[11] J. Tate , On Stark’s conjectures on the behavior of L(s,X) at s = 0 . J. Fac. Sci. U. Tokyo Sect. IA 28 (1981), No. 3, 963-978. · Zbl 0514.12013
[12] A. Weil , Jacobi sums as Grössencharaktere . Trans Am. Math. Soc. 23 (1952) 487-495. · Zbl 0048.27001 · doi:10.2307/1990804
[13] A. Weil , Sommes de Jacobi et caracteres de Hecke . Nachr. Akad. Wiss. Göttingen, Math.-Phys. Klasse (1974) 1-14. · Zbl 0367.10035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.