Lubin-Tate Groups Research Articles

Finiteness of analytic cohomology of Lubin-Tate (φL,ΓL)-modules

We prove finiteness and base change properties for analytic cohomology of families of L-analytic (φL,ΓL)-modules parametrised by affinoid algebras in the sense of Tate. For technical reasons we work over a field K containing a period of the Lubin-Tate group, which allows us to describe analytic cohomology in terms of an explicit generalised Herr complex.

Formal groups and lifts of the field of norms

Let $K$ be a finite extension of $\mathbf{Q}_p$. The field of norms of a strictly APF extension $K_\infty/K$ is a local field of characteristic $p$ equipped with an action of $\mathrm{Gal}(K_\infty/K)$. When can we lift this action to characteristic zero, along with a compatible Frobenius map ? In this article, we explain what we mean by lifting the field of norms, explain its relevance to the theory of $(\varphi,\Gamma)$-modules, and show that under a certain assumption on the type of lift, such an extension is generated by the torsion points of a relative Lubin-Tate group and that the power series giving the lift of the action of the Galois group of $K_\infty/K$ are twists of semi-conjugates of endomorphisms of the same relative Lubin-Tate group.

A representative of RΓ(N,T) for higher dimensional twists of ℤpr(1)

Let [Formula: see text] be a Galois extension of [Formula: see text]-adic number fields and let [Formula: see text] be a de Rham representation of the absolute Galois group [Formula: see text] of [Formula: see text]. In the case [Formula: see text], the equivariant local [Formula: see text]-constant conjecture describes the compatibility of the equivariant Tamagawa number conjecture with the functional equation of Artin [Formula: see text]-functions and it can be formulated as the vanishing of a certain element [Formula: see text] in [Formula: see text]; a similar approach can be followed also in the case of unramified twists [Formula: see text] of [Formula: see text] (see [W. Bley and A. Cobbe, The equivariant local [Formula: see text]-constant conjecture for unramified twists of [Formula: see text], Acta Arith. 178(4) (2017) 313–383; D. Izychev and O. Venjakob, Equivariant epsilon conjecture for 1-dimensional Lubin–Tate groups, J. Théor. Nr. Bordx. 28(2) (2016) 485–521]). One of the main technical difficulties in the computation of [Formula: see text] arises from the so-called cohomological term [Formula: see text], which requires the construction of a bounded complex [Formula: see text] of cohomologically trivial modules which represents [Formula: see text] for a full [Formula: see text]-stable [Formula: see text]-sublattice [Formula: see text] of [Formula: see text]. In this paper, we generalize the construction of [Formula: see text] in Theorem 2 of [W. Bley and A. Cobbe, The equivariant local [Formula: see text]-constant conjecture for unramified twists of [Formula: see text], Acta Arith. 178(4) (2017) 313–383] to the case of a higher dimensional [Formula: see text].

Formal Groups over Sub-Rings of the Ring of Integers of a Multidimensional Local Field

We constructed so-called convergence rings for the ring of integers of a multidimensional local field. The convergence ring is a sub-ring of the ring of integers with the property that any power series with coefficients from the sub-ring converges when replacing a variable by an arbitrary element of the maximal ideal. The properties of convergence rings and an explicit formula for their construction are derived. Note that the multidimensional case is fundamentally different from the case of the classical (one-dimensional) local field, where the whole ring of integers is the convergence ring. Next, we consider a multidimensional local field with zero characteristics of the penultimate residue field. For each convergence ring of such a field, we introduce a homomorphism that allows us to construct a formal group over the same ring with a logarithm having coefficients from the field for a power series with coefficients from the ring, and we give an explicit formula for the coefficients. In addition, by isogeny with coefficients from this ring, we construct a generalization of the formal Lubin—Tate group over this ring, study the endomorphisms of these formal groups, and derive a criterion for their isomorphism. We prove a one-to-one correspondence between formal groups created by ring homomorphism and by isogeny. Also, for any finite extension of a multidimensional local field with zero characteristic of the penultimate residue field, we consider the point group generated by the corresponding Lubin—Tate formal group.

Classification of Generalized Formal Lubin–Tate Groups over Multidimensional Local Fields

Classification of Generalized Formal Lubin–Tate Groups over Multidimensional Local Fields

Formal Modules for Relative Formal Lubin–Tate Groups

Relative formal Lubin–Tate groups are studied, namely, their structure, the ring of endomorphisms, and the group of points. The primary elements are considered, and an explicit formula for the generalized Hilbert symbol is derived.

Formal Modules for Generalized Lubin–Tate Groups

The structure, endomorphism ring, and point group of a generalized Lubin–Tate formal group are studied. The primary elements are examined and an explicit formula for the generalized Hilbert symbol is proved. Bibliography: 10 titles.

Iterated extensions and relative Lubin-Tate groups

Let K be a finite extension of \({\mathbf {Q}}_{p}\) with residue field \({\mathbf {F}}_q\) and let \(P(T) = T^d + a_{d-1}T^{d-1} + \cdots +a_1 T\) where d is a power of q and \(a_i \in {\mathfrak {m}}_K\) for all i. Let \(u_0\) be a uniformizer of \({\mathcal {O}}_K\) and let \(\{u_n\}_{n \geqslant 0}\) be a sequence of elements of \({\overline{\mathbf {Q}}}_{p}\) such that \(P(u_{n+1}) = u_n\) for all \(n \geqslant 0\). Let \(K_\infty \) be the field generated over K by all the \(u_n\). If \(K_\infty /K\) is a Galois extension, then it is abelian, and our main result is that it is generated by the torsion points of a relative Lubin-Tate group (a generalization of the usual Lubin-Tate groups). The proof of this involves generalizing the construction of Coleman power series, constructing some p-adic periods in Fontaine’s rings, and using local class field theory.

Equivariant epsilon conjecture for 1-dimensional Lubin-Tate groups

In this paper we formulate a conjecture on the relationship be- tween the equivariant -constants (associated to a local p-adic representation V and a nite extension of local elds L=K) and local Galois cohomology groups of a Galois stable Zp-lattice T of V. We prove the conjecture for L=K being an unramied extension of degree prime to p and T being a p-adic Tate module of a one-dimensional Lubin-Tate group dened over Zp by extending the ideas of (Breu) from the case of the multiplicative group Gm to arbi- trary one-dimensional Lubin-Tate groups. For the connection to the dierent formulations of the -conjecture in (BB), (FK), (Breu), (BlB) and (BF) see (Iz).

The Hilbert Symbol in Higher-Dimensional Local Fields for Formal Lubin–Tate Groups. II

The Hilbert Symbol in Higher-Dimensional Local Fields for Formal Lubin–Tate Groups. II

A note on Fontaine theory using different Lubin-Tate groups

The starting point of Fontaine theory is the possibility of translating the study of a p-adic representation of the absolute Galois group of a finite extension K of Qp into the investigation of a (φ, Γ)-module. This is done by decomposing the Galois group along a totally ramified extension of K, via the theory of the field of norms: the extension used is obtained by means of the cyclotomic tower which, in turn, is associated to the multiplicative Lubin-Tate group. It is known that one can insert different Lubin-Tate groups into the "Fontaine theory" machine to obtain equivalences with new categories of (φ, Γ)-modules (here φ may be iterated). This article uses only (φ, Γ)-theoretical terms to compare the different (φ, Γ) modules arising from various Lubin-Tate groups.

The Eisenstein reciprocity law for formal Lubin–Tate groups

A generalized norm residue symbol on Lubin–Tate formal groups is studied. The triviality of this symbol in the case where the first argument belongs to the definition field of the formal group is investigated. Explicit formulas for the generalized norm residue symbol ( , ) F,n are used. To this end, a restriction on the expansion of the first argument in powers of a uniformizer is removed. Bibliography: 6 titles.

Galois representations and Lubin-Tate groups

Using Lubin-Tate groups, we develop a variant of Fontaine's theory of (\varphi, \Gamma) -modules, and we use it to give a description of the Galois stable lattices inside certain crystalline representations.

Integer valued polynomials and Lubin–Tate formal groups

If R is an integral domain and K is its field of fractions, we let Int ( R ) stand for the subring of K [ x ] which maps R into itself. We show that if R is the ring of integers of a p-adic field, then Int ( R ) is generated, as an R-algebra, by the coefficients of the endomorphisms of any Lubin–Tate group attached to R.

Rank one solvable p-adic differential equations and finite Abelian characters via Lubin–Tate groups

We introduce a new class of exponentials of Artin–Hasse type, called π-exponentials. These exponentials depend on the choice of a generator π of the Tate module of a Lubin–Tate group \(\mathfrak{G}\) over \(\mathbb{Z}_p\). They arise naturally as solutions of solvable differential modules over the Robba ring. If \(\mathfrak{G}\) is isomorphic to \(\widehat{\mathbb{G}}_m\) over \(\mathbb{Z}_p\), we develop methods to test their over-convergence, and get in this way a stronger version of the Frobenius structure theorem for differential equations. We define a natural transformation of the Artin–Schreier complex into the Kummer complex. This provides an explicit generator of the Kummer unramified extension of \(\epsilon^\dagger_{K_{\infty}}\), whose residue field is a given Artin–Schreier extension of \(k(\!(t)\!)\), where k is the residue field of K. We then compute explicitly the group, under tensor product, of isomorphism classes of rank one solvable differential equations. Moreover, we get a canonical way to compute the rank one φ-module over \(\epsilon^\dagger_{K_\infty}\) attached to a rank one representation of \(\mathrm{Gal}(k(\!(t)\!)^{\mathrm{sep}}/k(\!(t)\!))\), defined by an Artin–Schreier character.

Explicit Reciprocity Law for Lubin–Tate Formal Groups

In this article, using Fontaine’s ΦΓ–module theory, we give a new proof of Coleman’s explicit reciprocity law, which generalizes that of Artin–Hasse, Iwasawa andWiles, by giving a complete formula for the norm residue symbol on Lubin–Tate groups. The method used here is different from the classical ones and can be used to study the Iwasawa theory of crystalline representations.

A simple proof of a lemma of Coleman

Let p be an odd prime. The results in this paper concern the units of the infinite extension of Qp generated by all p-power roots of unity. Letformula herewhere μpn+1 denote the pn+1th roots of 1. Let [pscr ]n be the maximal ideal of the ring of integers of Φn and let Un be the units congruent to 1 modulo [pscr ]n.Let ζn be a fixed primitive pn+1th root of unity such that ζpn = ζn − 1, ∀n [ges ] 1. Put πn = ζn − 1. Thus πn is a local parameter for Φn. Letformula hereKummer already exploited the obvious fact that every u0 ∈ U0 can be written in the formformula herewhere f0(T) is some power series in Zp[[T]]. Here of course, the power series f0(T) is not uniquely determined. Letformula herethe inverse limit being with respect to norm maps. Coates and Wiles (see [3]) discovered that any unit u = (un) ∈ U∞ has a unique power series fu(T) in Zp[[T]] with fu(πn) = un. The uniqueness of such a power series is obvious by Weierstrass Preparation Theorem, but the existence is in no way obvious. They worked with the formal group of height one attached to an elliptic curve with complex multiplication at an ordinary prime, but their ideas apply to any Lubin–Tate group defined over Zp. Almost immediately, Coleman [4] gave a totally different proof of the existence of the fu(T), which holds for all Lubin–Tate groups. We refer to such a power series as a Coleman power series. In this paper we adopt the same approach as [3]. We first prove the following result which is stronger than the original one in [3].

An explicit form of the Hilbert pairing for the relative formal Lubin-Tate groups

In the present paper an explicit formula is derived for the Hilbert symbol on the relative formal Lubin-Tate groups. This formula is a generalization of the Vostokov formula for the ordinary Lubin-Take groups. Bibliography:3 titles.

Norm property of Hilbert pairing

The natural analog of the property (x, 1 − x)=1, x ≠ 0,1 of the multiplicative Hilbert symbol is considered for Hilbert pairing in Lubin-Tate formal groups; in this paper the property is called the fundamental norm property. Criteria are given for satisfaction of the fundamental norm property in Lubin-Tate formal groups, and ϕ-symbols generalizing this property are described. A new approach to constructing formal Lubin-Tate groups is presented.

Explicit reciprocity laws on relative Lubin-Tate groups

Explicit reciprocity laws on relative Lubin-Tate groups