Integrality of Stickelberger elements and the equivariant Tamagawa number conjecture (Q323740)

This paper is an important contribution to the theory of generalized Stickelberger ideals and the Equivariant Tamagawa Conjecture (ETNC) in the noncommutative setting. Giving the statement of ETNC for a general \(G\)-Galois extension \(L/K\) of number fields and the appropriate motive would take up too much space; let us just say that its validity has nice arithmetical applications, e.g., in proving the non-abelian Brumer conjecture.NEWLINENEWLINEAs in the commutative case, Stickelberger elements attached to \(L/K\) (and some auxiliary parameters) are characterized by their character values in terms of \(L\)-values at 0. They always lie in the center of the group ring \(\mathbb Q[G]\). Experience shows that in order to construct an arithmetically meaningful (generalized) Stickelberger ideal, just one Stickelberger element will not suffice: to obtain the ``correct'' ideal, one also has to consider Stickelberger elements attached to subextensions \(L_J/K\) cut out by non-ramification conditions, where \(J\) ranges over all subsets of the ramification set Ram\((L/K)\), and pack the whole collection into an ideal \(SKu'(L/K)\), called the Stickelberger-Kurihara ideal. Its structure is rather complicated already in the abelian case. (Note: In the author's paper the definition of \(SKu'(L/K)\) is not given in terms of Stickelberger elements but via a certain equivariant \(L\)-value \(L(0)^{\#}\) and a module \(U'\) of Sinnott type. But exactly as in the abelian case, one can rewrite the definition in terms of (corestrictions of) Stickelberger elements, cf. for example Prop.~2.4 in the reviewer's paper [Compos. Math. 143, No. 6, 1399--1426 (2007; Zbl 1135.11059)]. We also note that before defining \(SKu'(L/K)\) (which he calls ``the modified S.-K. ideal''), the author introduces an ideal \(SKu(L/K)\), but this un-primed version does not seem to play a prominent role in this paper.)NEWLINENEWLINEHe formulates two integrality conjectures: one for Stickelberger elements, and one for \(SKu'(L/K)\). ``Integrality'' here means being contained in (the center of) the maximal order \(\mathcal M(G)\) of \(\mathbb Q[G]\). Under some standard technical hypotheses, he is able to prove both conjectures for monomial groups \(G\). A group is monomial if all of its irreducible characters over \(\mathbb C\) are induced from a linear character of some subgroup. Monomial groups form an interesting and fairly wide class, sitting between nilpotent and solvable groups. (Remarks: The smallest non-nilpotent monomial group is \(S_3\); the smallest non-monomial solvable group is \(\mathrm{SL}(2,3)\). The class of monomial groups seems to be the widest class for which Artin's conjecture on holomorphy of \(L\)-functions is proved.) It is interesting to note that in one of the results, groups of the form \(G=H\times C\) are considered with \(H\) monomial and \(C\) abelian. It is true that \(G\) is then monomial as well, but the point in splitting off the abelian factor \(C\) is that the author proves containment in \({\mathcal M}(H)[C]\), and usually the order \({\mathcal M}(H)[C]\) is considerably smaller than the order \({\mathcal M}(H\times C)\).NEWLINENEWLINEAnother main result of the paper is a proof of ETNC for monomial \(G\) under an appropriate \(\mu=0\) assumption, based on the integrality results just described. This is a very nice coping stone to the author's previous paper [Compos. Math. 147, No. 4, 1179--1204 (2011; Zbl 1276.11174)], and a kind of converse to earlier results of his. For reasons of space we refer the reader to Section 6.3 the paper for all details, and we also will not discuss the interesting applications given in \S\S7-8. The proof of the main result uses among other things a clever reduction to \(p\)-elementary groups. (For the reader's convenience let us recall that a group is \(p\)-elementary if it decomposes as a product of a \(p\)-group and a cyclic factor whose order is prime to \(p\).)