arXiv subjects
Publications and source records attributed to Andreas Nickel.
Let p be an odd prime. We give an unconditional proof of the equivariant Iwasawa main conjecture for totally real fields for an infinite class of one-dimensional non-abelian p-adic Lie extensions. Crucially, this result does not depend on the vanishing of the relevant Iwasawa mu-invariant.
Let L/K be a finite Galois extension of number fields with Galois group G. Let p be a rational prime and let r be a non-positive integer. By examining the structure of the p-adic group ring Z_p[G], we prove many new cases of the p-part of the equivariant Tamagawa number conjecture (ETNC) for the pair (h^0(Spec(L)(r),Z[G])). The same methods can also be applied to other conjectures concerning the vanishing of certain elements in relative algebraic K-groups. We then prove a conjecture of Burns concerning the annihilation of class groups as Galois modules for a wide class of interesting extensions, including cases in which the full ETNC in not known. Similarly, we construct annihilators of higher dimensional algebraic K-groups of the ring of integers in L.
Let $\mathcal O$ be the ring of integers in a finite extension of $\mathbb Q_p$. If $G$ is a finite group and $Γ$ is a maximal order containing the group ring $\mathcal O[G]$, Jacobinski's conductor formula gives a complete description of the central conductor of $Γ$ into $\mathcal O[G]$ in terms of characters of $G$. We prove a similar result for completed group algebras $\mathcal O[[G]]$, where $G$ is a $p$-adic Lie group of dimension $1$. We will also discuss several consequences of this result.
Let $L/K$ be a finite Galois CM-extension of number fields with Galois group $G$. In an earlier paper, the author has defined a module $SKu(L/K)$ over the center of the group ring $\mathbb Z[G]$ which coincides with the Sinnott-Kurihara ideal if $G$ is abelian and, in particular, contains many Stickelberger elements. It was shown that a certain conjecture on the integrality of $SKu(L/K)$ implies the minus part of the equivariant Tamagawa number conjecture at an odd prime $p$ for an infinite class of (non-abelian) Galois CM-extensions of number fields which are at most tamely ramified above $p$, provided that Iwasawa's $μ$-invariant vanishes. Here, we prove a relevant part of this integrality conjecture which enables us to deduce the equivariant Tamagawa number conjecture from the vanishing of $μ$ for the same class of extensions. As an application we prove the non-abelian Brumer and Brumer-Stark conjecture outside the $2$-primary part for any monomial Galois extension of the rationals provided that certain $μ$-invariants vanish.
To each finitely presented module M over a commutative ring R one can associate an R-ideal Fit_R(M) which is called the (zeroth) Fitting ideal of M over R and which is always contained in the R-annihilator of M. In an earlier article, the second author generalised this notion by replacing R with a (not necessarily commutative) o-order Lambda in a finite dimensional separable algebra, where o is an integrally closed complete commutative noetherian local domain. To obtain annihilators, one has to multiply the Fitting invariant of a (left) Lambda-module M by a certain ideal H(Lambda) of the centre of Lambda. In contrast to the commutative case, this ideal can be properly contained in the centre of Lambda. In the present article, we determine explicit lower bounds for H(Lambda) in many cases. Furthermore, we define a class of `nice' orders Lambda over which Fitting invariants have several useful properties such as good behaviour with respect to direct sums of modules, computability in a certain sense, and H(Lambda) being the best possible.
We discuss three different formulations of the equivariant Iwasawa main conjecture attached to an extension K/k of totally real fields with Galois group G, where k is a number field and G is a p-adic Lie group of dimension 1 for an odd prime p. All these formulations are equivalent and hold if Iwasawa's μ-invariant vanishes. Under mild hypotheses, we use this to prove non-abelian generalizations of Brumer's conjecture, the Brumer-Stark conjecture and a strong version of the Coates-Sinnott conjecture provided that μ= 0.