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Motivic \(L\)-functions and Galois module structures. (English) Zbl 0867.11081

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 J.-M. Fontaine and B. Perrin-Riou [Proc. Symp. Pure Math. 55, 599–706 (1994; Zbl 0821.14013)] and, independently, by 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 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 A. Fröhlich [J. Reine Angew. Math. 397, 42–99 (1989; Zbl 0693.12012)] as well as a conjecture of Chinburg are established.
Reviewer: G.Lettl (Graz)

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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References:

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