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Twisted \(S\)-units, \(p\)-adic class number formulas, and the Lichtenbaum conjectures. (English) Zbl 0863.19003
This is an important paper in which delicate methods from Galois cohomology and Iwasawa theory are used, in conjunction with deep results in algebraic K-theory, to establish a version of the Lichtenbaum conjectures for absolutely abelian fields \(F\). These conjectures are far-reaching generalizations of the analytic class number formula, which will be recalled just below. Before we enter into the details of the paper under review, we’ll try to give a general impression of what is going on. The Lichtenbaum conjectures are concerned with predictions about the order of even-numbered \(K\)-groups of the ring of integers \({\mathcal O}_F\) in a number field \(F\), and about regulators attached to odd-numbered \(K\)-groups. The philosophy is that even-numbered K-groups behave like ideal class groups, and the odd-numbered ones behave like unit groups. At the very least, one knows that \(K_{2i}({\mathcal O}_F)\) is finite, and \(K_{2i-1}({\mathcal O}_F)\) is finitely generated with known rank, by results of A. Borel; \(K_0({\mathcal O}_F)\) is the direct sum of the ideal class group of \(F\) and a copy of \(\mathbb{Z}\), and \(K_1({\mathcal O}_F)\) is exactly the unit group of \({\mathcal O}_F\). Now the analytic class number formula provides information about the class group of \(F\), and the regulator of \(F\), which is a quantity attached to the unit group of \({\mathcal O}_F\). Actually, when one uses the functional equation of the Dedekind zeta function \(\zeta_F\), the more usual version of the analytic class number formula, involving the residue at \(s=1\), changes into the neater statement \(\zeta_F(0)^*=-h_FR_F/w_F\), where \(h_F\) and \(R_F\) are the class number and the regulator of \(F\), \(w_F\) counts the roots of unity in \(F\), and * means: take the first nonvanishing coefficient in the Taylor expansion at \(s=0\). This can be taken to be the Lichtenbaum conjecture for \(m=1\) (of course, it is a long-standing theorem).
Let us now briefly recall the Lichtenbaum conjecture \(LC_m\) for \(m\geq2\) and explain in the process what is proved in the paper under review. Briefly, the leading coefficient of the development of \(\zeta_F(s)\) at \(s=1-m\) is conjectured to be given as a quotient \[ \zeta_F(1-m)^*={|K_{2m-2}({\mathcal O}_F)|\over w_m(F)}\cdot R_m(F)\cdot r, \] where \(K_*\) is Quillen’s \(K\)-theory, the number \(w_m(F)\) is a straightforward generalization of \(w_F=w_1(F)\), the number of roots of unity in \(F\), \(R_m(F)\) is a certain regulator attached to \(K_{2m-1}({\mathcal O}_F)\) (it is a determinant as the usual regulator, but we will not attempt to explain it), and \(r\) is another rational factor.
The main result of paper under review proves the above equality for \(F\) abelian over \(\mathbb Q\), with the following specifications and modifications: \(r\) is a certain explicit product of Euler factors, times a (nonexplicit) power of 2. Furthermore, the Quillen \(K\)-group \(K_{2m-2}({\mathcal O}_F)\) is replaced by a group \(K^g_{2m-2}({\mathcal O}_F)\) which needs to be explained: For every prime \(p\), there is an epimorphism \(ch_{2,m}: K_{2m-2}({\mathcal O}_F)\otimes\mathbb Z_p \to K_{2m-2}^{\text{ét}}({\mathcal O}_F)\), and its target (called étale K-theory; the notation suppresses the dependency on \(p\)) is isomorphic to the étale cohomology group H\(^2({\mathcal O}_F,\mathbb Z_p(m))=\text{H}^2_{\text{ét}}({\mathcal O}_F[{1/p}],\mathbb Z_p(m))\). This is due to Dwyer and Friedlander. (The étale cohomology groups just mentioned can also be expressed by Galois cohomology.) Now the group \(K^g_{2m-2}({\mathcal O}_F)\) is the direct sum over all \(p\) of the image of a splitting of \(ch_{2,m}\), which exists by work of B. Kahn. So the main result of the paper under review basically replaces \(K\)-theory by étale \(K\)-theory. Another way of looking at this is the following: The kernel of \(ch_{2,m}\) is finite and expected to be zero for odd \(p\); this is the so-called Quillen-Lichtenbaum conjecture, which is known to hold for \(m=2\). One may therefore say: the Lichtenbaum conjecture is proved modulo the Quillen-Lichtenbaum conjecture. (There is a quite recent preprint of B. Kahn containing a proof of the 2-primary part of the Quillen-Lichtenbaum conjecture.) We should also mention that the Chern class maps \(ch_{2,m}\) have analogs \(ch_{1,m}: K_{2m-1}({\mathcal O}_F)\otimes\mathbb{Z}_p \to K_{2m-1}^{\text{ét}}({\mathcal O}_F)\); they are again onto, with finite kernel, and the kernel is again conjectured to be zero (neglecting the 2-part). However, the regulator “does not see” the unknown but finite kernel of \(ch_{1,m}\).
The long and clever argument begins with the identification of étale K-theory with certain cohomology groups H\(^i({\mathcal O}_F,\mathbb{Z}_p(m))\), with \(i=1\) and \(i=2\). It has been accepted for some time that these cohomology groups should behave like “twisted” unit groups resp. class groups. Precise results in this direction are proved in §3. To wit, there is an exact sequence \[ 0 \to {\bar U}'_\infty(m-1)_{G_\infty}\to\text{H}^1({\mathcal O}_F,\mathbb{Z}_p(m))\to X'_\infty(m-1)^{G_\infty}\to 0. \] The \({\bar U'}\) term is obtained by taking a projective limit of groups of \(p\)-units in a \(p\)-cyclotomic extension, twisting \(m-1\) times, and “descending back” by taking coinvariants. The H\(^1\) term is an étale \(K\)-group, and the cokernel \(X'_\infty(m-1)^{G_\infty}\) (which comes from Iwasawa theory) is finite. One might think of this as a twisted version of the result which states that \(K_1\) of a ring of integers is just the unit group. The authors also introduce \({\bar C}'_\infty\), a subgroup of \({\bar U}'_\infty\) constructed from cyclotomic units, which will be crucial in the sequel.
The principal theorem of this paper on the algebraic side (§5) is a sharpened version of the following index formula (we disregard the character decomposition to keep things simpler): \[ [\text{H}^1({\mathcal O}_F,\mathbb Z_p(m))^\pm : {\bar C}'_\infty(m-1)_{G_\infty} ^\pm]=|\text{H}^2({\mathcal O}_F,\mathbb Z_p(m))^\pm| \] with plus signs for \(m\) odd and minus signs for \(m\) even. This is an analog of Kummer’s result describing the plus class number \(h_\ell^+\) as an index of cyclotomic units inside a full unit group. The proof of the index formula just mentioned combines intricate cohomological calculations with Iwasawa theory in a very nice way, and it relies on most of the preceding results. The main theorem of the paper, which has already been explained above, is subsequently deduced in §6 with the help of this theorem, using a lot of deep results of Beilinson, Bloch and Soulé on regulator maps and cyclotomic elements in \(K\)-theory.
Two small criticisms: The proof of the remark on p. 692 (which is very important for §5) is incorrect as it stands. The authors are aware of the problem; a corrigendum is in preparation and should appear in due course. Second point: On p.715, a certain element \(C_m^D(w)\) is called “Galois generator” of a certain group \(\tilde C_\infty(m-1)_{G_\infty}\). (In the fifth line: read \(\zeta_{p^n}\) for \(\zeta_{pn}\).) I’m not sure whether the group in question is generated by that single element; perhaps one should allow \(w\) to vary over all \(N\)th roots of unity. The reader is referred without a page or theorem number to P. Deligne’s fundamental paper [in: Galois groups over \(\mathbb Q\), Publ., Math. Sci. Res. Inst. 16, 79-297 (1989; Zbl 0742.14022)] which is quite long.
The last remarks should be considered as minor matters. In the reviewer’s opinion, this is a paper which cleverly employs a lot of powerful techniques and proves a major result.
[This review has also been sent to Math. Rev.]

MSC:
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
11R42 Zeta functions and \(L\)-functions of number fields
11R34 Galois cohomology
11R23 Iwasawa theory
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