35 research outputs found

    A noncommutative Weierstrass preparation theorem and applications to Iwasawa theory

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    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

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    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

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    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

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    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

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    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

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    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 pp-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

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    For the pp-cyclotomic tower of Qp\mathbb{Q}_p Fontaine established a description of local Iwasawa cohomology with coefficients in a local Galois representation VV in terms of the ψ\psi-operator acting on the attached etale (Ο†,Ξ“)(\varphi,\Gamma)-module D(V)D(V). In this article we generalize Fontaine's result to the case of arbitratry Lubin-Tate towers L∞L_\infty over finite extensions LL of Qp\mathbb{Q}_p by using the Kisin-Ren/Fontaine equivalence of categories between Galois representations and (Ο†L,Ξ“L)(\varphi_L,\Gamma_L)-module and extending parts of [Herr L.: Sur la cohomologie galoisienne des corps pp-adiques. Bull. Soc. Math. France 126, 563-600 (1998)], [Scholl A. J.: Higher fields of norms and (Ο•,Ξ“)(\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∞L_\infty for the multiplicative group twisted with the dual of the Tate module TT of the Lubin-Tate formal group in terms of Coleman power series and the attached (Ο†L,Ξ“L)(\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.: LL-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(Ο‡LTr)L(\chi_{LT}^r) in terms of generalized Coates-Wiles homomorphisms, where the Lubin-Tate characater Ο‡LT\chi_{LT} describes the Galois action on T.T.Comment: 54 page

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

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    In \citenospec{MR1097615} several spectral sequences for (global and local) Iwasawa modules over (not necessarily commutative) Iwasawa algebras (mainly of pp-adic Lie groups) over Zp\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 Zp\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

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    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

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    In arXiv:math/0404297 a non-commutative Iwasawa Main Conjecture for elliptic curves over Q\mathbb{Q} has been formulated. In this note we show that it holds for all CM-elliptic curves EE defined over Q\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) pp-adic LL-function of EE 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) pp-adic LL-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
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