Noncommutative Fitting invariants and improved annihilation results

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.Comment: 24 pages; appendix deleted, many corrections and improvements following referee's report. To appear in J. Lond. Math. So

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

We show that for an odd prime p, the p-primary parts of refinements of the (imprimitive) non-abelian Brumer and Brumer-Stark conjectures are implied by the equivariant Iwasawa main conjecture (EIMC) for totally real fields. Crucially, this result does not depend on the vanishing of the relevant Iwasawa mu-invariant. In combination with the authors' previous work on the EIMC, this leads to unconditional proofs of the non-abelian Brumer and Brumer-Stark conjectures in many new cases.Comment: 33 pages; to appear in Mathematische Zeitschrift; v3 many minor updates including new title; v2 some cohomological arguments simplified; v1 is a revised version of the second half of arXiv:1408.4934v

On the equivariant Tamagawa number conjecture for Tate motives and unconditional annihilation results

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.Comment: 33 pages, error in section 3.4 corrected. To appear in Transactions of the AM

Hybrid Iwasawa algebras and the equivariant Iwasawa main conjecture

This is the author accepted manuscript. It is currently pending publication by Johns Hopkins University Press - to appear in the American Journal of Mathematics.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 µ-invariant.It is a pleasure to thank Werner Bley, Ted Chinburg, Takako Fukaya, Lennart Gehrmann, Cornelius Greither, Annette Huber-Klawitter, Mahesh Kakde, Kazuya Kato, Daniel Macias Castillo, Cristian Popescu, J¨urgen Ritter, Sujatha, Otmar Venjakob, Christopher Voll, Al Weiss and Malte Witte for helpful discussions and correspondence. The authors also thank the referee for several helpful comments. The second named author acknowledges financial support provided by the DFG within the Collaborative Research Center 701 ‘Spectral Structures and Topological Methods in Mathematics’

On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions

Let A be an abelian variety defined over a number field k and let F be a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we compute explicitly the algebraic part of the p-component of the equivariant Tamagawa number of the pair (h1(A/F)(1),Z[Gal(F/k)]). By comparing the result of this computation with the theorem of Gross and Zagier we are able to give the first verification of the p-component of the equivariant Tamagawa number conjecture for an abelian variety in the technically most demanding case in which the relevant Mordell–Weil group has strictly positive rank and the relevant field extension is both non-abelian and of degree divisible by p. More generally, our approach leads us to the formulation of certain precise families of conjectural p-adic congruences between the values at s = 1 of derivatives of the Hasse–Weil L-functions associated to twists of A, normalised by a product of explicit equivariant regulators and periods, and to explicit predictions concerning the Galois structure of Tate–Shafarevich groups. In several interesting cases we provide theoretical and numerical evidence in support of these more general predictions