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On the equivariant Tamagawa number conjecture for Tate motives. (English) Zbl 1142.11076
To a “motif” $$M$$ (given by its cohomological realizations) are attached a complex function $$L(M, s)$$ and all the data conjecturally describing the leading Taylor coefficient $$L^\ast (M)$$ of this function at $$s = 0.$$ Very roughly speaking, Beilinson’s conjecture describes $$L^\ast (M) \in {\mathbb R}^\ast$$ in terms of regulators up to a rational factor, and the Bloch-Kato conjecture gives this rational factor in terms of so-called Tamagawa numbers (commutative algebraic groups being replaced by motivic cohomology groups). A later reformulation by Fontaine and Perrin-Riou of this Tamagawa number conjecture in terms of complexes and determinants could be viewed (roughly speaking again) as some kind of very elaborate global-local principle bringing together the archimedean and $$p$$-adic worlds. Finally, the equivariant version (ETNC for short) proposed by Burns and Flach provides a coherent overview and refinement of many existing conjectures on the arithmetical interpretation of special values of $$L$$-functions in connection with Galois module structures.
In this paper, the authors prove the ETNC for Tate motives, more specifically for pairs $$(h^0 (\text{Spec} (L)) (r), {\mathbb Z} [{1 \over 2}] [\text{Gal} (L/K)]),$$ $$r$$ being a negative or null integer, $$L$$ a finite abelian number field, $$K$$ any subfield of $$L.$$ The proof can be divided in two parts:
1) Go up the cyclotomic tower $${\mathbb Q}(\xi_{mp^\infty}), p \not= 2, p\nmid m,$$ and prove a kind of “equivariant Iwasawa main conjecture” formulated by Kato in terms of complexes (thm. 6-1). This combines a systematic use of the Iwasawa theory of perfect complexes (in the spirit e.g. of Nekovar) together with more classical apparatus such as the theorem of Mazur-Wiles and results on $$\mu$$-invariants.
2) Prove the $$p$$-part of the ETNC (for the motif under consideration) by Iwasawa descent on complexes. In the case $$r < 0,$$ descent is “classical” but relies heavily on fundamental results of Beilinson-Huber-Wildeshaus relating special values of Dirichlet $$L$$-functions to polylogarithms and cyclotomic elements in higher algebraic $$K$$-theory. In the case $$r = 0,$$ which is notoriously more delicate due to the presence of “trivial zeroes” of $$p$$-adic $$L$$-functions, crucial use is made of results of Ferrero-Greenberg-Gross on the first derivative of $$p$$-adic $$L$$-functions at $$s = 0$$ and valuative properties of certain canonical “cyclotomic $$p$$-units” shown by Solomon (§ 9). Note related work (but weaker results) by D. Benois and T. Nguyen-Quang-Do [Ann. Sci. Éc. Norm. Supér. (4) 35, 641–672 (2002; Zbl 1125.11351)] on the (original) Bloch-Kato conjecture for Tate motives over an abelian number field, and by A. Huber and G. Kings [Duke Math. J. 119, 393–464 (2003; Zbl 1044.11095)] for Dirichlet motives. Note also that the case $$p = 2$$ has been dealt with by [M. Flach, Stark’s conjectures: recent work and new directions. Papers from the international conference on Stark’s conjectures and related topics, Johns Hopkins University, Baltimore, MD, USA, August 5–9, 2002. Providence, RI: American Mathematical Society (AMS). Contemporary Mathematics 358, 79–125 (2004; Zbl 1070.11025)].

##### MSC:
 11R23 Iwasawa theory 11R18 Cyclotomic extensions
##### Keywords:
equivariant; complexes; determinants
##### Citations:
Zbl 1125.11351; Zbl 1044.11095; Zbl 1070.11025
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