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The Brumer–Stark conjecture is a conjecture in algebraic number theory giving a rough generalization of both the analytic class number formula for Dedekind zeta functions, and also of Stickelberger's theorem about the factorization of Gauss sums. It is named after Armand Brumer and Harold Stark. It arises as a special case (abelian and first-order) of Stark's conjecture, when the place that splits completely in the extension is finite. There are very few cases where the conjecture is known to be valid. Its importance arises, for instance, from its connection with Hilbert's twelfth problem. == Statement of the conjecture == Let be an abelian extension of global fields, and let be a set of places of containing the Archimedean places and the prime ideals that ramify in . The -imprimitive equivariant Artin L-function is obtained from the usual equivariant Artin L-function by removing the Euler factors corresponding to the primes in from the Artin L-functions from which the equivariant function is built. It is a function on the complex numbers taking values in the complex group ring where is the Galois group of . It is analytic on the entire plane, excepting a lone simple pole at . Let be the group of roots of unity in . The group acts on ; let be the annihilator of as a -module. An important theorem, first proved by C. L. Siegel and later independently by Takuro Shintani, states that is actually in . A deeper theorem, proved independently by Pierre Deligne and Ken Ribet, Daniel Barsky, and Pierrette Cassou-Noguès, states that is in . In particular, is in , where is the cardinality of . The ideal class group of is a . From the above discussion, we can let act on it. The Brumer–Stark conjecture says the following: Brumer–Stark Conjecture. For each nonzero fractional ideal of , there is an "anti-unit" such that # #The extension is abelian. The first part of this conjecture is due to Armand Brumer, and Harold Stark originally suggested that the second condition might hold. The conjecture was first stated in published form by John Tate.〔Tate, John, (Brumer–Stark–Stickelberger ), Séminaire de Théorie des Nombres, Univ. Bordeaux I Talence, (1980-81), exposé no. 24.〕 The term "anti-unit" refers to the condition that is required to be 1 for each Archimedean place .〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Brumer–Stark conjecture」の詳細全文を読む スポンサード リンク
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