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Determining Fitting ideals of minus class groups via the equivariant Tamagawa number conjecture. (English) Zbl 1135.11059

In this paper \(K/k\) is an abelian number field extension with Galois group \(G\) and with \(k\) totally real and \(K\) a CM field; \(c\in G\) is the complex conjugation on \(K\) and \(\text{cl}^-_K= \mathbb{Z}[{1\over 2}][G]/(1+ c)\bigotimes_{\mathbb{Z}[G]}\text{cl}_K\) the non 2-part of the minus class group of \(K\). Moreover, throughout the paper the equivariant Tamagawa number conjecture (ETNC) is assumed to hold for the motive \(h^0(K)\) with coefficients in \(\mathbb{Z}[G]\) (which is the case if \(k= \mathbb{Q}\) [D. Burns and C. Greither, Invent. Math. 153, No. 2, 303–359 (2003; Zbl 1142.11076)]). The so-called Sinnott-Kurihara ideal \(SKu(K/k)\) is an important tool in determining the (initial) Fitting ideal \(\text{Fitt}(\text{cl}^-_K)\) of \(\text{cl}^-_K\). It is derived from the element \(\omega\in\mathbb{Q}[G]\) satisfying \(\chi(\omega)= L(0,\breve\chi)\) for the characters \(\chi\) of \(G\) (compare [W. Sinnott, Invent. Math. 62, 181–234 (1980; Zbl 0465.12001)]) and a generalized version of Stickelberger’s ideal involving certain local modules \(U_{{\mathfrak p}}\subset\mathbb{Q}[G_{{\mathfrak p}}]\) which, in turn, are related to the duals of the K. W. Gruenberg and A. Weiss inertia modules \(W_{{\mathfrak p}}\) [Q. J. Math., Oxf. II. Ser. 47, No. 185, 25–39 (1996; Zbl 0858.11060)] appearing in the construction of Tate sequences \(E_S\to A\to B\to\nabla_S\) for small sets \(S\) [J. Ritter and A. Weiss, Compos. Math. 102, No. 2, 147–178 (1996; Zbl 0948.11041)] (it is just the set \(S_\infty\) of all Archimedean places that, in these terms, is connected with the class group \(\text{cl}_K\) rather than, for general \(S\), with the \(S\)-class group \(\text{cl}_{K,S}= \text{tor}(\nabla_S)\)).
Theorem: \(SKu(K/k)^-\) is the Fitting ideal of the Pontryagin dual of \(\text{cl}^-_K\), provided that the group of odd-order roots of unity in \(K\) has projective dimension \(\leq 1\).
Very roughly speaking, the point here is this. When \(S\) is large, the Tate sequence and a Dirichlet map \(\Delta S= \nabla S\to E_S\) give rise to an element in the relative \(K\)-group \(K_0T(\mathbb{Z}[G]\simeq K_0(\mathbb{Z}[G], \mathbb{Q})\) [K. W. Gruenberg, J. Ritter and A. Weiss, Proc. Lond. Math. Soc. (3) 79, No. 1, 47–80 (1999; Zbl 1041.11075)] as well as to a refined Euler characteristic [D. Burns, Compos. Math. 129, No. 2, 203–237 (2001; Zbl 1014.11070)] which, assuming ETNC, is closely related to the image of the equivariant \(L\)-value \(L^*_S(0)^\sharp\) at zero under \(\partial: K_1(\mathbb{R}[G])\to K_0(\mathbb{Z}[{1\over 2}][G]/(1 +c),\mathbb{R})\), and which moreover can be linked to \(\partial\) applied to the Fitting ideal of \((\nabla_{S_\infty}/\delta(C))^-\) for some natural \(\delta: C= \mathbb{Z}[G][S\setminus S_\infty]\to \nabla_S\). The paper finishes with a corollary saying that Brumer’s conjecture holds outside the 2-part.
Related work is [C. Greither, Math. Z. 233, No. 3, 515–534 (2000; Zbl 0965.11047); M. Kurihara, J. Reine Angew. Math. 561, 39–86 (2003; Zbl 1056.11063) and Doc. Math., J. DMV Extra Vol., 539–563 (2003; Zbl 1135.11339)].

MSC:

11R29 Class numbers, class groups, discriminants
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R42 Zeta functions and \(L\)-functions of number fields
19B28 \(K_1\) of group rings and orders
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