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**The equivariant Tamagawa number conjecture: a survey.**
*(English)*
Zbl 1070.11025

Burns, David (ed.) et al., 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) (ISBN 0-8218-3480-0/pbk). Contemporary Mathematics 358, 79-125 (2004).

This article is an expanded version of a survey talk given at the conference on Stark’s conjecture held in Baltimore in August 2002. In part 1, the author recalls the Tamagawa Number (ala Bloch-Kato) conjecture in the formulation of Fontaine and Perrin-Riou. In part 2 (resp. 3), he gives its equivariant version (ETNC for short) with commutative (resp. noncommutative) coefficients. The importance of the ETNC could hardly be underestimated. For its detailed expression and whereabouts, we refer to our review of the paper of A. Huber and G. Kings [Duke Math. J. 119, No. 3, 393–464 (2003; Zbl 1044.11095)]. Let us just say here that the ETNC aims to give an arithmetic (\(p\)-adic) interpretation of special values of L-functions attached to motives, in a very general setting which encompasses various and profound conjectures in number theory and arithmetic geometry such as the Lichtenbaum conjecture on special values of zeta functions of number fields, the Birch and Swinnerton-Dyer conjecture for elliptic curves, the conjectures of Stark, Chinburg and others on the Galois module structure of \(K\)-groups (including units) of number fields…

The author puts a particular emphasis on proven cases. A detailed outline of the proof of the Burns-Greither theorem is given, including the case \(p=2.\) Recall that D. Burns and C. Greither [Invent. Math. 153, 303–359 (2003; Zbl 1142.11076)] proved the ETNC for Tate motives attached to absolutely abelian number fields, for \(p\) odd. Their method is Iwasawa theoretic, which means that they first prove, at infinite level, an Equivariant Main Conjecture (EMC for short) formulated in terms of complexes, then they prove the ETNC by descent. One key technical step is the vanishing of the \(\mu\) invariant attached to global units modulo circular units, which is proved in an appendix by C. Greither (including the case \(p=2\)).

(Reviewer’s remark: the Burns-Greither theorem has been recently extended by W. Bley (preprint, 2005) to abelian extensions of an imaginary quadratic field in which \(p\) splits). This kind or Iwasawa theoretic approach should also work for motives \(M(\psi)\) associated to algebraic Hecke characters of an imaginary quadratic field: the EMC is actually simpler, but our knowledge of motivic cohomology for the motives \(M(\psi)\) is not yet sufficient. Particular cases of the ETNC have nevertheless been proved, e.g. by J. A. Colwell [The Conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication by a nonmaximal order (thesis, 2003)] in a “critical” case, and by G. Kings [Invent. Math. 143, 571–627 (2001; Zbl 1159.11311)] in a non-critical case. The situation is the same for motives \(M(f)\) attached to modular newforms \(f\), although the results obtained so far are even less complete. Note that a strategy for proving an Iwasawa EMC for the cyclotomic deformation of motives \(M(f)\) at ordinary primes \(p\) has been recently outlined by Skinner and Urban, and descent computations should allow to deduce the \(p\)-part of the Birch and Swinnerton-Dyer conjecture if the order of the zero of the L-function at \(s=1\) is at most 1. In the same vein, H. Darmon and M. Bertolini [to appear in Ann. Math. (2)] study the EMC for the anti-cyclotomic deformation of motives M(f) with respect to an auxiliary quadratic field and an ordinary prime \(p\). A different approach, not based on Iwasawa theory but on the Taylor-Wiles method developed to show the modularity of elliptic curves over the rationals, has proved to be successful for some adjoint motives of modular forms (see e.g. H. G. Diamond, M. Flach and Guo, Ann. Sci. Éc. Norm. Supér. (4) 37, No. 5, 663–727 (2004; Zbl 1121.11045), or Quiang Lin (thesis, 2003).

All in all, this is a very clear, complete and precise survey, with a comprehensive bibliography of 87 references, many of them being preprints or articles of the 21st century.

For the entire collection see [Zbl 1052.11003].

The author puts a particular emphasis on proven cases. A detailed outline of the proof of the Burns-Greither theorem is given, including the case \(p=2.\) Recall that D. Burns and C. Greither [Invent. Math. 153, 303–359 (2003; Zbl 1142.11076)] proved the ETNC for Tate motives attached to absolutely abelian number fields, for \(p\) odd. Their method is Iwasawa theoretic, which means that they first prove, at infinite level, an Equivariant Main Conjecture (EMC for short) formulated in terms of complexes, then they prove the ETNC by descent. One key technical step is the vanishing of the \(\mu\) invariant attached to global units modulo circular units, which is proved in an appendix by C. Greither (including the case \(p=2\)).

(Reviewer’s remark: the Burns-Greither theorem has been recently extended by W. Bley (preprint, 2005) to abelian extensions of an imaginary quadratic field in which \(p\) splits). This kind or Iwasawa theoretic approach should also work for motives \(M(\psi)\) associated to algebraic Hecke characters of an imaginary quadratic field: the EMC is actually simpler, but our knowledge of motivic cohomology for the motives \(M(\psi)\) is not yet sufficient. Particular cases of the ETNC have nevertheless been proved, e.g. by J. A. Colwell [The Conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication by a nonmaximal order (thesis, 2003)] in a “critical” case, and by G. Kings [Invent. Math. 143, 571–627 (2001; Zbl 1159.11311)] in a non-critical case. The situation is the same for motives \(M(f)\) attached to modular newforms \(f\), although the results obtained so far are even less complete. Note that a strategy for proving an Iwasawa EMC for the cyclotomic deformation of motives \(M(f)\) at ordinary primes \(p\) has been recently outlined by Skinner and Urban, and descent computations should allow to deduce the \(p\)-part of the Birch and Swinnerton-Dyer conjecture if the order of the zero of the L-function at \(s=1\) is at most 1. In the same vein, H. Darmon and M. Bertolini [to appear in Ann. Math. (2)] study the EMC for the anti-cyclotomic deformation of motives M(f) with respect to an auxiliary quadratic field and an ordinary prime \(p\). A different approach, not based on Iwasawa theory but on the Taylor-Wiles method developed to show the modularity of elliptic curves over the rationals, has proved to be successful for some adjoint motives of modular forms (see e.g. H. G. Diamond, M. Flach and Guo, Ann. Sci. Éc. Norm. Supér. (4) 37, No. 5, 663–727 (2004; Zbl 1121.11045), or Quiang Lin (thesis, 2003).

All in all, this is a very clear, complete and precise survey, with a comprehensive bibliography of 87 references, many of them being preprints or articles of the 21st century.

For the entire collection see [Zbl 1052.11003].

Reviewer: Thong Nguyen Quang Do (Besançon)