On higher order Stickelberger-type theorems
Let \(L/K\) be an abelian extension of global fields with Galois group \(G\). In the paper under review the author formulates an explicit refinement of a conjecture of \textit{K. Rubin} [Ann. Inst. Fourier 46, No. 1, 33--62 (1996; Zbl 0834.11044)]. Assuming his conjecture, he derives several explicit consequences on the annihilation of (modifications of) class groups as a Galois module. This is in the spirit of classical conjectures as Brumer's conjecture on the annihilation of class groups. In this case, the conjectural annihilators (`Stickelberger elements') are defined via values at zero of Artin \(L\)-functions of the irreducible characters of \(G\). However, it happens frequently that these Stickelberger elements vanish so that the conjecture does not predict anything. More recently, some authors began to use leading terms rather than values of Artin \(L\)-series to construct conjectural annihilators of the class group; see for instance the universal approach of \textit{D. Burns} [Invent. Math. 186, No. 2, 291--371 (2011; Zbl 1239.11128)]. In this article the author assumes the validity of the equivariant Tamagawa number conjecture (ETNC) as formulated by \textit{D. Burns} and \textit{M. Flach} [Doc. Math., J. DMV 6, 501--570 (2001; Zbl 1052.11077)] resp.~its function field analogue to construct conjectural annihilators via higher derivatives of abelian Artin \(L\)-functions. As a consequence he obtains unconditional results in many interesting cases. Namely, the function field analogue of the ETNC is known by a results of \textit{D. Burns} [Invent. Math. 184, No. 2, 221--256 (2011; Zbl 1234.11080)]; in the case of number fields, the ETNC is known for absolutely abelian extensions by \textit{D. Burns} and \textit{C. Greither} [Invent. Math. 153, No. 2, 303--359 (2003; Zbl 1142.11076)]. Finally, the author derives several results for abelian extensions of imaginary quadratic fields building on work of \textit{W. Bley} [Doc. Math., J. DMV 11, 73--118 (2006; Zbl 1178.11070)]. Reviewer's remark: In the latter case the author excludes the case \(p=3\) as Bley uses a result of \textit{R. Gillard} [J. Reine Angew. Math. 358, 76--91 (1985; Zbl 0551.12011)] on the vanishing of certain \(\mu\)-invariants. However, this case has very recently been settled by \textit{H.~Oukhaba} and \textit{S.~Viguié} [``On the \(\mu\)-invariant of Katz \(p\)-adic \(L\) functions attached to imaginary quadratic fields and applications'', Preprint, \url{arXiv:1311.3565}].
scientific article; zbMATH DE number 6341791
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