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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)].

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)].

Reviewer: Thong Nguyen Quang Do (Besançon)