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From the Birch and Swinnerton-Dyer conjecture to non-commutative Iwasawa theory via the equivariant Tamagawa number conjecture – a survey. (English) Zbl 1145.11076

Burns, David (ed.) et al., \(L\)-functions and Galois representations. Based on the symposium, Durham, UK, July 19–30, 2004. Cambridge: Cambridge University Press (ISBN 978-0-521-69415-5/pbk). London Mathematical Society Lecture Note Series 320, 333-380 (2007).
This paper aims to give a survey on T. Fukaya and K. Kato’s article [Uraltseva, N.N.(ed.), Proceedings of the St. Petersburg Mathematical Society. Vol. XII. Translations. Series 2. Am. Math. Soc. 219, 1–85 (2006; Zbl 1238.11105)] which establishes the relation between the equivariant Tamagawa number conjecture (ETNC) of D. Burns and M. Flach [Doc. Math. J. DMV 6, 501–570 (2001; Zbl 1052.11077)] and the noncommutative Iwasawa main conjecture (IMC) with \(p\)-adic \(L\) functions as formulated e.g. by J. Coates, T. Fukaya, K. Kato, R. Sujatha and O. Venjakob [Publ. Math., Inst. Hautes Étud. Sci. 101, 163–208 (2005; Zbl 1108.11081)], and accessorily the IMC without \(p\)-adic \(L\)-functions as formulated by A. Huber and G. Kings [in: Li, Ta Tsien (ed.) et al., Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20–28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press, 149–162 (2002; Zbl 1020.11067)]. The subject is still evolving, and it is anyway too vast for us to give anything else but a fugitive glimpse. On the one hand, the absolute TNC was first proposed by Bloch and Kato as a far reaching generalization of the class number formula and the Birch and Swinnerton-Dyer conjecture. While the Deligne-Beilinson conjecture links the order of vanishing at \(s =0\) of the \(L\) function attached to a motive to its motivic cohomology and claims rationality of special \(L\)-values or more general leading coefficients (up to periods and regulators), the Bloch-Kato conjecture predicts the precise \(L\)-values in terms of Galois cohomology. The later formulation given by Fontaine and Perrin-Riou replaced (Tamagawa) measures by (commutative) determinants, and this is the approach which prevails now. In Kato’s view, when allowing other (commutative) coefficients than number fields, classical Iwasawa theory, roughly sepaking, can be considered as an ETNC for a “big” coefficient ring. Building on this insight, Burns and Flach formulated an ETCN where the coefficients of the motive are allowed to be (noncommutative) finite-dimensional \({\mathbb Q}\)-algebras, using as general tools the (noncommutative) determinant functor and relative algebraic \(K\)-groups. Their systematic approach recovers all previous versions of the TNC and moreover all central conjectures of Galois module theory (conjectures “à la” Stark, “à la” Chinburg, etc \(\dots\)).
On the other hand, noncommutative Iwasawa theory studies the arithmetic of Galois extensions whose Galois group are \(p\)-adic Lie groups (with or without \(p\)-torsion), generalizing the classical study of \({\mathbb Z}_p\)-extensions. At least in the so called “ordinary good reduction” case, the now prevailing formulation of the IMC uses a localization exact sequence for \(K\)-groups in dimensions 0,1 : roughly speaking, a conjectural \(p\)-adic \({\mathfrak L}\)-function (which must satisfy certain \(p\)-adic interpolation properties) should live in a “new localized \(K_1\)” and should map by the connecting homomorphism to the class of a certain module or complex which lives in a relative \(K_0\)-group. Of course, modulo an adequate translation, this contains the classical IMC for \({\mathbb Z}_p\)-extensions (Wiles’ theorem). Note that in recent work by Ritter-Weiss, Hara, Kakde, etc \(\ldots\), special (but general enough) cases of the IMC have been proved when the \(p\)-adic Lie group is of dimension 1.
In Fukaya-Kato’s attempt to relate the ETNC and the IMC, it is sufficient to use noncommutative coefficients only for the Galois cohomology, but to stick to number fields as coefficients for the involved motives. In this expository paper, the author follows closely this approach, concentrating on the case of an abelian variety (although setting the stage for general motives) over \({\mathbb Q}\) and \(p\)-adic Lie extensions of \({\mathbb Q}.\) In this context, the TNC can be extended to an equivariant version using the absolute version for all twists of the motive by certain representations of the \(p\)-adic Lie group. Interested readers are urged to consult further this well-written and very useful survey.
For the entire collection see [Zbl 1130.11004].

MSC:

11R23 Iwasawa theory
11-02 Research exposition (monographs, survey articles) pertaining to number theory
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