Motivic \(L\)-functions and Galois module structures
During the last decades, many results about varieties over global fields were (at least conjecturally) generalized to motives, which today occupy a prominent position in arithmetic algebraic geometry. Using perfect complexes and their determinants, the conjectures of Bloch and Kato about \(L\)-functions were recently extended to motives with coefficients by \textit{J.-M. Fontaine} and \textit{B. Perrin-Riou} [Proc. Symp. Pure Math. 55, 599--706 (1994; Zbl 0821.14013)] and, independently, by \textit{K. Kato} [Kodai Math. J. 16, 1--31 (1993; Zbl 0798.11050)]. The paper under review discusses applications of these extended conjectures to Galois module theory, namely by defining invariants which describe the Galois module structure of various cohomology groups arising from motives defined over an algebraic number field and admitting an action of a finite abelian Galois group. In the case of Tate motives of arbitrary weight, the existence of these invariants is established independently of conjectures in Proposition 1.42, and connections to Chinburg's invariants, as introduced by \textit{T. Chinburg} [Invent. Math. 74, 321--349 (1983; Zbl 0564.12016) and Ann. Math. (2) 121, 351--376 (1985; Zbl 0567.12010)], are given in Proposition 1.48. In the last section, relations between conjectures about the value of motivic \(L\)-functions at zero and results of \textit{A. Fröhlich} [J. Reine Angew. Math. 397, 42--99 (1989; Zbl 0693.12012)] as well as a conjecture of Chinburg are established.
value of motivic \(L\)-functions at zero
scientific article; zbMATH DE number 892579
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