Searcharxiv⌕ Search

arXiv subjects

Otmar Venjakob

Publications and source records attributed to Otmar Venjakob.

At least 19 recordsLinked to original sources
arXiv2025

$ε$-isomorphisms for rank one $(φ,Γ)$-modules over Lubin-Tate Robba rings

Inspired by Nakamura's work (arXiv:1305.0880) on $ε$-isomorphisms for $(φ,Γ)$-modules over (relative) Robba rings with respect to the cyclotomic theory, we formulate an analogous conjecture for $L$-analytic Lubin-Tate $(φ_L,Γ_L)$-modules over (relative) Robba rings for any finite extension $L$ of $\mathbb{Q}_p.$ In contrast to Kato's and Nakamura's setting, our conjecture involves $L$-analytic cohomology instead of continuous cohomology within the generalized Herr complex. Similarly, we restrict to the identity components of $D_{cris}$ and $D_{dR},$ respectively. For rank one modules of the above type or slightly more generally for trianguline ones, we construct $ε$-isomorphisms for their Lubin-Tate deformations satisfying the desired interpolation property.

math.NT↗
arXiv2024

On Lubin-Tate regulator maps and Kato's explicit reciprocity law

We extend the interpolation property of the Lubin-Tate regulator map from [SV24] to Artin characters and show a reciprocity law in the sense of Cherbonnier-Colmez. This allows us to provide a new proof of Kato's explicit reciprocity law for Lubin-Tate formal groups.

math.NT↗
arXiv2023

Reciprocity laws for $(φ_L,Γ_L)$-modules over Lubin-Tate extensions

In the Lubin-Tate setting we study pairings for analytic $(φ_L,Γ_L)$-modules and prove an abstract reciprocity law which then implies a relation between the analogue of Perrin-Riou's Big Exponential map as developed by Berger and Fourquaux and a $p$-adic regulator map whose construction relies on the theory of Kisin-Ren modules generalising the concept of Wach modules to the Lubin-Tate situation.

math.NT↗
arXiv2023

Compairing categories of Lubin-Tate $(φ_L,Γ_L)$-modules

In the Lubin-Tate setting we compare different categories of $(φ_L,Γ_L)$-modules over various perfect or imperfect coefficient rings. Moreover, we study their associated Herr-complexes. Finally, we show that a Lubin Tate extension gives rise to a weakly decompleting, but not decompleting tower in the sense of Kedlaya and Liu.

math.NT↗
arXiv2020

Herr-complexes in the Lubin-Tate setting

In this article we extend work of Herr from the case of cyclotomic $(φ,Γ)$-modules to the general case of Lubin-Tate $(φ,Γ)$-modules. In particular, we define generalized $φ$- and $ψ$-Herr complexes, which calculate Galois cohomology, when applied to the etale $(φ,Γ)$-modules attached to the coefficients.

math.NT↗
arXiv2017

On Spectral Sequences for Iwasawa Adjoints à 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 à la Grothendieck (and Tate for duality groups).

math.NT↗
arXiv2017

Wach modules, regulator maps, and epsilon-isomorphisms in families

We prove the local epsilon-isomorphism conjecture of Fukaya and Kato [FK06] for certain crystalline families of G_Qp-representations. This conjecture can be regarded as a local analogue of the Iwasawa main conjecture for families. Our work extends earlier work of Kato for rank-1 modules (cf. [Ven13]), of Benois and Berger for crystalline G_Qp-representations with respect to the cyclotomic extension (cf. [BB08]), as well as of Loeffler, Venjakob, and Zerbes (cf. [LVZ13]) for crystalline G_Qp- representations with respect to abelian p-adic Lie extensions of Qp. Nakamura [Nak13, Nak14] has also formulated a version of the epsilon-conjecture for affinoid families of (phi,Gamma)-modules over the Robba ring, and proved his conjecture in the rank-1 case. He used this case to construct an epsilon-isomorphism for families of trianguline (phi,Gamma)-modules, depending on a fixed triangulation. Our results imply that this epsilon-isomorphism is independent of the chosen triangulation for certain crystalline families. The main ingredient of our proof consists of the construction of families of Wach modules generalizing work of Wach and Berger [Ber04] and following the approach of Kisin to the construction of potentially semi-stable deformation rings [Kisa].

math.NT↗
arXiv2015

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 $ψ$-operator acting on the attached etale $(φ,Γ)$-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 $(φ_L,Γ_L)$-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 $(ϕ,Γ)$-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 $(φ_L,Γ_L)$-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, Birkhäuser Boston 1990] Thm. 2.1 to our situation expressing the Bloch-Kato exponential map for $L(χ_{LT}^r)$ in terms of generalized Coates-Wiles homomorphisms, where the Lubin-Tate characater $χ_{LT}$ describes the Galois action on $T.$

math.NT↗
arXiv2013

Equivariant epsilon conjecture for 1-dimensional Lubin-Tate groups

In this paper we formulate a conjecture on the relationship between the equivariant ε-constants (associated to a local p-adic representation V and a finite extension of local fields L/K) and local Galois cohomology groups of a Galois stable \mathbb{Z}_{p}-lattice T of V. We prove the conjecture for L/K being an unramified extension of degree prime to p and T being a p-adic Tate module of a one-dimensional Lubin-Tate group defined over \mathbb{Z}_{p} by extending the ideas of \cite{Breu} from the case of the multiplicative group \mathbb{G}_{m} to arbitrary one-dimensional Lubin-Tate groups. For the connection to the different formulations of the ε-conjecture in \cite{BB}, \cite{FK}, \cite{Breu}, \cite{BlB} and \cite{BF} see \cite{Iz}.

math.NT↗
arXiv2013

On Kato's local epsilon-isomorphism Conjecture for rank one Iwasawa modules

This paper contains a complete proof of Fukaya's and Kato's epsilon$-isomorphism conjecture in [23] for invertible Λ-modules (the case of V = V_0(r) where V_0 is unramified of dimension 1). Our results rely heavily on Kato's unpublished proof of (commutative) epsilon-isomorphisms for one dimensional representations of G_{Q_p} in [27], but apart from fixing some sign-ambiguities in (loc.\ cit.) we use the theory of (ϕ,Γ)-modules instead of syntomic cohomology. Also, for the convenience of the reader we give a slight modification or rather reformulation of it in the language of [23] and extend it to the (slightly non-commutative) semi-global setting. Finally we discuss some direct applications concerning the Iwasawa theory of CM elliptic curves, in particular the local Iwasawa Main Conjecture for CM elliptic curves E over the extension of Q_p which trivialises the p-power division points E(p) of E. In this sense the paper is complimentary to the joint work [7] on noncommutative Main Conjectures for CM elliptic curves.

math.NT↗
arXiv2013

Local epsilon isomorphisms

In this paper, we prove the "local epsilon-isomorphism conjecture" of Fukaya and Kato for a particular class of Galois modules obtained by tensoring a Zp-lattice in a crystalline representation of the Galois group of Qp with a representation of an abelian quotient of the Galois group with values in a suitable p-adic local ring. This can be regarded as a local analogue of the Iwasawa main conjecture for abelian p-adic Lie extensions of Qp, extending earlier work of Benois and Berger for the cyclotomic extension. We show that such an epsilon-isomorphism can be constructed using the Perrin-Riou regulator map, or its extension to the 2-variable case due to the first and third authors.

math.NT↗
arXiv2011

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.

math.KT↗
arXiv2011

On the work of Ritter and Weiss in Comparison with Kakde's Approach

Almost simultaneously Ritter and Weiss arXiv:1004.2578 on the one hand and Kakde arXiv:1008.0142 on the other hand gave a proof of the non-commutative Iwasawa main conjecture over totally real fields for the Tate motive under the assumption that a certain $μ$-invariant vanishes as has been conjectured also by Iwasawa. In this notes we compare both approaches in a survey.

math.NT↗
arXiv2010

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.

math.NT↗
arXiv2010

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.

math.KT↗
arXiv2007

Localisations and Completions of Skew Power Series Rings

This paper is a natural continuation of the study of skew power series rings A initiated in [P. Schneider and O. Venjakob, On the codimension of modules over skew power series rings with applications to Iwasawa algebras, J. Pure Appl. Algebra 204 (2005), 349 - 367.]. We construct skew Laurent series rings B and show the existence of some canonical Ore sets S for the skew power series rings A such that a certain completion of the localisation A_S is isomorphic to B. This is applied to certain Iwasawa algebras. Finally we introduce subrings of overconvergent skew Laurent series rings.

math.RA↗