On the non-abelian Brumer-Stark conjecture and the equivariant Iwasawa main conjecture (Q2312836)

Let \(L/K\) be a finite Galois CM-extension, that is, \(K\) is a totally real number field, \(L\) is a CM field and \(L/K\) is a finite Galois extension. Set \(G:=\mathrm{Gal}(L/K)\). Consider, for each finite set \(S\) containing all the archimedean places and the ramified places, the Stickelberger element \(\theta_S(L/K)\). Let \(\mu_L\) and \(\mathrm{cl}_L\) denote the set of roots of unity and the class group of \(L\), respectively. Suppose \(G\) is abelian. The Brumer's conjecture asserts that \(\mathrm{Ann}_{{\mathbb Z}[G]}(\mu_L)\theta_S(L/K)\subseteq \mathrm{Ann}_{{\mathbb Z}[G]}(\mathrm{cl}_L)\). The Brumer-Stark conjecture asserts that the class of a given ideal becomes principal in \(L\) and gives information about a generator of that ideal. The non-abelian Brumer and Brumer-Stark conjectures are presented in Section 3: Conjectures 3.2 and 3.6. The main results, Theorem 5.2 and its Corollary 5.4, say that for an odd prime \(p\), the relevant case of the equivariant Iwasawa main conjecture for totally real fields implies the \(p\)-primary part of a dual version of the generalized non-abelian Brumer-Stark conjecture under the hypothesis that \(S\) contains the \(p\)-adic places of \(K\) and that a certain identity between complex and \(p\)-adic Artin \(L\)-functions at \(s=0\) holds. This result does not depend on the vanishing of the relevant Iwasawa \(\mu\)-invariant. Furthermore, in Section 10, in combination with previous results on the equivariant Iwasawa main conjecture, unconditional proofs of the non-abelian Brumer and Brumer-Stark conjectures are given in several new cases. The proof of the main results is the content of Sections 6, 7 and 8.