The main goal of this paper is to develop an explicit descent theory in the context of Whitehead groups of non-commutative Iwasawa algebras. Several explicit consequences of a ‘main conjecture’ for the values at \(s=1\) of twisted Hasse–Weil \(L\)-functions, of the kind formulated by J. Coates, T. Fukaya, K. Kato, R. Sujatha and the second named author in [Publ. Math., Inst. Hautes Étud. Sci. 101, 163–208 (2005; Zbl 1108.11081)], have been obtained by Coates et al. [loc. cit.], by K. Kato [\(K\)–Theory 34, No. 2, 99–140 (2005; Zbl 1080.19002)] and by T. Dokchitser and V. Dokchitser [Proc. Lond. Math. Soc. (3) 94, No. 1, 211–272 (2007; Zbl 1206.11083)]. However all of these consequences become trivial whenever the \(L\)–functions vanish at \(s=1\). Further, the conjecture of Birch and Swinnerton–Dyer together with a recent result of B. Mazur and K. Rubin [J. Differ. Geom. 70, No. 1, 1–22 (2005; Zbl 1211.11068)] imply that these \(L\) functions should vanish often. Therefore it is interesting to understand what a main conjecture predicts concerning the values of derivatives of Hasse–Weil \(L\)–functions at \(s=1\).
In this article the authors give the first step towards developing such a theory by obtaining a general formalism of descent theory of non-commutative Iwasawa theory. In a subsequent paper they will derive from the main conjecture of non-commutative Iwasawa theory several explicit and highly non-trivial congruence relations between values of derivatives of twisted Hasse–Weil \(L\)-functions. The descent theory developed here has a key role in obtaining the first verification of the equivariant Tamagawa number conjecture for a class of non-abelian extensions of number fields and in the proof of a conjecture of Chinburg in the setting of global function fields.
The paper is organized as follows. In §1 several preliminaries on localization of Iwasawa algebras, \(K\)-theory and derived categories are given. §2 contains the statement of the main \(K\)-theoretical results proved in this article. The first one is a decomposition theorem for Whitehead groups which is a natural generalization of the classical Weierstrass Preparation Theorem. The second one is the descent formula. This result recovers for an special case the classical descent formula discussed in [L. C. Washington, Introduction to cyclotomic fields. Graduate Texts in Mathematics, 83. New York-Heidelberg-Berlin: Springer-Verlag (1982; Zbl 0484.12001)], Example 13.12 and upon appropriated specialization recovers the descent formalism proved in certain commutative cases. In §3 it is defined a suitable notion of \(\mu\)–invariant and in §4 it is given a canonical ‘characteristic series’ in non–commutative Iwasawa theory. The canonical series is used in §5 to prove a formula for the ‘leading terms at Artin representations’ of elements of Whitehead groups of non-commutative Iwasawa algebras. In §6 it is shown that the main result of §5 is a satisfactory resolution of the descent problem. In §7 the authors formulate explicit main conjectures of non-commutative Iwasawa theory for both Tate motives and certain critical motives. In §8 the results of §5 are used to prove that, under suitable hypothesis, the main conjectures formulated in §7 imply the relevant special cases of the equivariant Tamagawa number conjecture formulated by M. Flach and the first named author in [Doc. Math., J. DMV 6, 501–570 (2001; Zbl 1052.11077)].