35 research outputs found

### A noncommutative Weierstrass preparation theorem and applications to Iwasawa theory

In this paper and a forthcoming joint one with Y. Hachimori we study Iwasawa
modules over an infinite Galois extension K of a number field k whose Galois
group G=G(K/k) is isomorphic to the semidirect product of two copies of the
p-adic numbers. After first analyzing some general algebraic properties of the
corresponding Iwasawa algebra, we apply these results to the Galois group of
the p-Hilbert class field over K.
As a main tool we prove a Weierstrass preparation theorem for certain skew
power series rings. One striking result in our work is the discovery of the
abundance of faithful torsion modules, i.e. non-trivial torsion modules whose
global annihilator ideal is zero. Finally we show that the completed group
algebra with coefficients in the finite field of p elements is a unique
factorization domain in the sense of Chatters

### Characteristic elements in noncommutative Iwasawa theory

In this article we construct characteristic elements for a certain class of
Iwasawa modules in noncommutative Iwasawa theory. These elements live in the
first K-group K_1(L_T) of the localisation L_T of the Iwasawa algebra L=L(G) of
a p-adic Lie group G with respect to a certain Ore-Set T. The evaluation of the
characteristic element of a module M under the Iwasawa algebra of the p-adic
Lie group G is related to the (twisted) G-Euler characteristic of M. We apply
these results to study the arithmetic of elliptic curves E (without CM) defined
over a number field k in the tower K=k(E(p)) of fields which arises by
adjoining the p-power division points to k. In particular, we relate the
characteristic element of the Selmer group of E over K, i.e. the algebraic
p-adic L-function of E over K, to the (classical) characteristic polynomial
associated with the Selmer group over the cyclotomic Z_p-extension. Finally we
discuss how the formulation of a noncommutative main conjecture could look like
assuming the existence of an analytic p-adic L-function

### From the Birch & Swinnerton-Dyer Conjecture over the Equivariant Tamagawa Number Conjecture to non-commutative Iwasawa theory - a survey

We give a survey on the relationship between non-commutative Iwasawa theory
and the Equivariant Tamagawa Number Conjecture following the work of (in
alphabetical order) Burns/Flach, Coates/Sujatha, Fukaya/Kato, Huber/Kings and
others.Comment: 36 page

### On the Iwasawa theory of p-adic Lie extensions

In this paper the new techniques and results concerning the structure theory
of modules over non-commutative Iwasawa algebras are applied to arithmetic: we
study Iwasawa modules over p-adic Lie extensions K of number fields k "up to
pseudo-isomorphism". In particular, a close relationship is revealed between
the Selmer group of abelian varieties, the Galois group of the maximal abelian
unramified p-extension of K as well as the Galois group of the maximal abelian
outside S unramified p-extension where S is a finite set of certain places of
k. Moreover, we determine the Galois module structure of local units and other
modules arising from Galois cohomology

### On the dimension theory of skew power series rings

The first purpose of this paper is to set up a general notion of skew power
series rings S over a coefficient ring R, which are then studied by filtered
ring techniques. The second subject consists of investigating the class of
S-modules which are finitely generated as R-module. In the case that S and R
are Auslander regular we show in particular that the codimension of M as
S-module is one higher than the codimension of M as R-module. Furthermore its
class in the Grothendieck group of S-modules of codimension at most one less
vanishes, which is in the spirit of the Gersten conjecture for commutative
regular local rings. Finally we apply these results to Iwasawa algebras of
p-adic Lie groups.Comment: 20 page

### K_1 of certain Iwasawa algebras, after Kakde

This paper contains a detailed exposition of the content of section five in
Kakde's paper arXiv:1008.0142. We proceed in a slightly more axiomatic way to
pin down the exact requirements on the $p$-adic Lie group under consideration.
We also make use of our conceptual theory of the completed localization of an
Iwasawa algebra as developed in arXiv:0711.2669. This simplifies some of the
arguments. Otherwise, with the exception of the notation at certain places, we
follow Kakde's paper.Comment: These are the notes of the "Instructional workshop on the
noncommutative main conjectures" held in M\"unster, April 26 - April 30, 2011
http://wwwmath.uni-muenster.de/u/schneider/Workshop_2011/index.htm

### Coates-Wiles homomorphisms and Iwasawa cohomology for Lubin-Tate extensions

For the $p$-cyclotomic tower of $\mathbb{Q}_p$ Fontaine established a
description of local Iwasawa cohomology with coefficients in a local Galois
representation $V$ in terms of the $\psi$-operator acting on the attached etale
$(\varphi,\Gamma)$-module $D(V)$. In this article we generalize Fontaine's
result to the case of arbitratry Lubin-Tate towers $L_\infty$ over finite
extensions $L$ of $\mathbb{Q}_p$ by using the Kisin-Ren/Fontaine equivalence of
categories between Galois representations and $(\varphi_L,\Gamma_L)$-module and
extending parts of [Herr L.: Sur la cohomologie galoisienne des corps
$p$-adiques. Bull. Soc. Math. France 126, 563-600 (1998)], [Scholl A. J.:
Higher fields of norms and $(\phi,\Gamma)$-modules. Documenta Math.\ 2006,
Extra Vol., 685-709]. Moreover, we prove a kind of explicit reciprocity law
which calculates the Kummer map over $L_\infty$ for the multiplicative group
twisted with the dual of the Tate module $T$ of the Lubin-Tate formal group in
terms of Coleman power series and the attached $(\varphi_L,\Gamma_L)$-module.
The proof is based on a generalized Schmid-Witt residue formula. Finally, we
extend the explicit reciprocity law of Bloch and Kato [Bloch S., Kato K.:
$L$-functions and Tamagawa numbers of motives. The Grothendieck Festschrift,
Vol. I, 333-400, Progress Math., 86, Birkh\"auser Boston 1990] Thm. 2.1 to our
situation expressing the Bloch-Kato exponential map for $L(\chi_{LT}^r)$ in
terms of generalized Coates-Wiles homomorphisms, where the Lubin-Tate
characater $\chi_{LT}$ describes the Galois action on $T.$Comment: 54 page

### On Spectral Sequences for Iwasawa Adjoints \`a la Jannsen for Families

In \citenospec{MR1097615} several spectral sequences for (global and local)
Iwasawa modules over (not necessarily commutative) Iwasawa algebras (mainly of
$p$-adic Lie groups) over $\Z_p$ are established, which are very useful for
determining certain properties of such modules in arithmetic applications.
Slight generalizations of said results can be found in \citenospec{MR2333680}
(for abelian groups and more general coefficient rings), \citenospec{MR1924402}
(for products of not necessarily abelian groups, but with $\Z_p$-coefficients),
and \citenospec{MR3084561}. Unfortunately, some of Jannsen's spectral sequences
for families of representations as coefficients for (local) Iwasawa cohomology
are still missing. We explain and follow the philosophy that all these spectral
sequences are consequences or analogues of local cohomology and duality \`a la
Grothendieck (and Tate for duality groups)

### A splitting for K_1 of completed group rings

Motivated by the theory of Coleman power series (reinterpreted via fields of
norms by Fontaine) we construct a splitting of the natural map of K_1 groups
arising from the mod p reduction map of the Iwasawa algebra of a pro-p Lie
group. We also show the vanishing of SK_1 for certain unipotent groups.Comment: 24 page

### On the non-commutative Main Conjecture for elliptic curves with complex multiplication

In arXiv:math/0404297 a non-commutative Iwasawa Main Conjecture for elliptic
curves over $\mathbb{Q}$ has been formulated. In this note we show that it
holds for all CM-elliptic curves $E$ defined over $\mathbb{Q}$. This was
claimed in (loc.\ cit.) without proof, which we want to provide now assuming
that the torsion conjecture holds in this case. Based on this we show firstly
the existence of the (non-commutative) $p$-adic $L$-function of $E$ and
secondly that the (non-commutative) Main Conjecture follows from the existence
of the Katz-measure, the work of Yager and Rubin's proof of the 2-variable main
conjecture. The main issues are the comparison of the involved periods and to
show that the (non-commutative) $p$-adic $L$-function is defined over the
conjectured in (loc.\ cit.) coefficient ring. Moreover we generalize our
considerations to the case of CM-elliptic cusp forms.Comment: 31 page