Euler systems for higher \(K\)-theory of number fields
The general notion of Euler systems attached to number fields was introduced by Kolyvagin, and special Euler systems of cyclotomic elements or Gauss sums, etc. \(\dots\) were used by Kolyvagin, Rubin and others to prove some important finiteness theorems in algebraic number theory. In Compos. Math. 81, No. 2, 223-236 (1992; Zbl 0747.11055), \textit{M. Kurihara} showed that the Deligne-Soulé cyclotomic elements \(c (\eta)\), where \(\eta\) is a root of unity of order not divisible by the prime number \(p\), constitute Euler systems in the étale cohomology groups \(H^1_{\text{ét}} (\mathbb{Z} [1/p, \eta], \mathbb{Z}_p (n + 1))\), and he used them to prove the finiteness of the groups \(H^2_{\text{ét}} (\mathbb{Z} [1/p],\;\mathbb{Z}_p (n + 1))\), for even \(n \geq 2\) (independently of Borel's results on \(K\)-groups; but note that Kurihara's proof, as published, is not quite correct). In a similar way, \textit{K. Kato} has recently constructed Euler systems for \(K_2\) of modular curves [Kodai Math. J. 16, No. 1, 1-31 (1993; Zbl 0798.11050)]. In the present paper, the authors construct Euler systems in higher odd \(K\)-groups of number fields. Fix an odd prime number \(p\) and a natural integer \(m\). Let \(S\) be the set of square free natural integers \(L\) whose prime factors split completely in a number field \(F\) and are congruent to 1 modulo \(p^m\). Let \(\{\alpha (L) \in F (\mu_{p^mL})\); \(L \in S\}\) be an Euler system in the sense of \textit{K. Rubin} [J. Reine Angew. Math. 425, 141-154 (1992; Zbl 0752.11045)]. Multiplying by special Bott elements and taking transfer from \(F (\mu_{p^m L})\) to \(F (\mu_L)\), the authors construct an Euler system of elements \(\wedge (L)\) living in \(K_{2n + 1} (F (\mu_L), \mathbb{Z}/p^m)\). Choosing interesting \(\alpha (L)\), one gets interesting Euler systems in \(K\)-theory and applications to number theory: 1) Taking \(\alpha (L)\) to be special cyclotomic elements and applying the Dwyer-Friedlander map, one recovers the system constructed by Kurihara. 2) Taking \(\alpha (L)\) to be specific Gauss sums which appear in Iwasawa theory of \(p\)-adic \(L\)-functions and applying Kolyvagin's operator \(D_L\), one gets elements \(K (L, \overline r) \in K_{2n + 1} (\mathbb{Z} [1/Lpr], \mathbb{Z}/p^m)\). Here \(\overline r\) is an ideal of \(\mathbb{Q} (\mu_{p^mL})\) over a prime number \(r\). 3) Applying the Dwyer-Friedlander map to the above \(K(L, \overline r)\) and using their functorial properties to perform a descent ``à la Kolyvagin'', the authors show that the order of \(H^2_{\text{ét}}(\mathbb{Z}[1/p],\;\mathbb{Z}_p(n+1))\), \(n\) odd, divides the \(p\)-part of \(w_{n + 1} (\mathbb{Q}) \zeta_\mathbb{Q} (-n)\). Note that the Main Theorem of Iwasawa theory actually implies the equality of the two quantities: this is the ``easy'' part of the Lichtenbaum conjecture for \(\mathbb{Q}\), or more generally for totally real number fields \(F\) (``easy'' as far as, for \(n\) odd and \(F\) totally real, no regulator is involved). In a terminal section -- which does not use at all Euler systems -- the authors give a formula for the order of the group \(D_{n + 1} (F)\) \((n\) even) of divisible elements in the \(p\)-primary part of \(K_{2n} (F)\), where \(F\) is an abelian totally real number field in which \(p\) does not ramify. For a general number field \(F\), the group \(D_{n + 1} (F)\) may be considered as a higher analogue of the wild kernel [see the first author, Compos. Math. 86, No. 3, 281-305 (1993; Zbl 0778.11066)], or independently, the reviewer, Sémin. Théor. Nombres Bordx., Sér. II 4, No. 2, 263-271 (1992; Zbl 0783.11042)]. Here the order of \(D_{n + 1} (F)\) is given as the index inside the group \(K^{\text{ét}}_{2n} (F)\) of the subgroup generated by Soulé's cyclotomic elmements. This is the higher analogue of the classical formula giving the ``plus'' part of the class number of cyclotomic fields as the index of circular units inside units. In the authors' mind, it completes their former result on the order of the group \(H^2_{\text{ét}} (\mathbb{Z} [1/p], \mathbb{Z}_p (n + 1))\), which is known to be actually isomorphic to \(D_{n + 1} (\mathbb{Q})\). But note that this is slightly misleading, the ``genuine'' index formula (for general abelian number fields \(F)\) involves in fact \(K^{\text{ét}}_{2n} ({\mathcal O}_F)\) instead of \(D_{n + 1} (F)\). See \textit{M. Kolster}, the reviewer and \textit{V. Fleckinger} [``Twisted \(S\)-units, \(p\)-adic class number formulas and the Lichtenbaum conjectures'', Duke Math. J. 84, No. 3, 679-717 (1996)], where the index formula is a key step towards their proof of the Lichtenbaum conjecture for abelian fields.
scientific article; zbMATH DE number 894605