One-dimensional Formal Group Research Articles

(G,χϕ)-equivariant ϕ-coordinated quasi modules for vertex algebras

To give a unified treatment on the association of Lie algebras and vertex algebras, we study (G,χϕ)-equivariant ϕ-coordinated quasi modules for vertex algebras, where G is a group with χϕ a linear character of G and ϕ is an associate of the one-dimensional additive formal group. The theory of (G,χϕ)-equivariant ϕ-coordinated quasi modules for nonlocal vertex algebra is established in [10]. In this paper, we concentrate on the context of vertex algebras. We establish several conceptual results, including a generalized commutator formula and a general construction of vertex algebras and their (G,χϕ)-equivariant ϕ-coordinated quasi modules. Furthermore, for any conformal algebra C, we construct a class of Lie algebras Cˆϕ[G] and prove that restricted Cˆϕ[G]-modules are exactly (G,χϕ)-equivariant ϕ-coordinated quasi modules for the universal enveloping vertex algebra of C. As an application, we determine the (G,χϕ)-equivariant ϕ-coordinated quasi modules for affine and Virasoro vertex algebras.

Lubin–Tate Deformation Spaces and Fields of Norms

We construct a tower of fields from the rings R n which parametrize pairs (X,λ), where X is a deformation of a fixed one-dimensional formal group 𝕏 of finite height h, together with a Drinfeld level-n structure λ. We choose principal prime ideals 𝔭 n |(p) in each ring R n in a compatible way and consider the field K n ′ obtained by localizing R n at 𝔭 n and passing to the field of fractions of the completion. By taking the compositum K n =K n ′ K 0 of K n ′ with the completion K 0 of a certain unramified extension of K 0 ′ , we obtain a tower of fields (K n ) n which we prove to be strictly deeply ramified in the sense of Scholl. When h=2 we also investigate the question of whether this is a Kummer tower.

The norm residue symbol for higher local fields

In this paper we investigate the Kummer pairing associated to an arbitrary (one-dimensional) formal group. In particular, we obtain formulae describing the values of the pairing in terms of multidimensional p-adic differentiation, the logarithm of the formal group, the generalized trace and the norm on Milnor K-groups. The results are a generalization to higher-dimensional local fields of Kolyvagin's explicit reciprocity laws. In particular, they constitute a generalization of the reciprocity laws of Artin-Hasse, Iwasawa and Wiles.

On the quantum affine vertex algebra associated with trigonometric R-matrix

We apply the theory of \(\phi \)-coordinated modules, developed by H.-S. Li, to the Etingof–Kazhdan quantum affine vertex algebra associated with the trigonometric R-matrix of type A. We prove, for a certain associate \(\phi \) of the one-dimensional additive formal group, that any (irreducible) \(\phi \)-coordinated module for the level \(c\in {\mathbb {C}}\) quantum affine vertex algebra is naturally equipped with a structure of (irreducible) restricted level c module for the quantum affine algebra in type A and vice versa. In the end, we discuss relation between the centers of the quantum affine algebra and the quantum affine vertex algebra.

The Hecke algebra action and the Rezk logarithm on Morava E-theory of height 2

Given a one-dimensional formal group of height 2, let E be the Morava E-theory spectrum associated to its universal deformation over the Lubin-Tate ring. By computing with moduli spaces of elliptic curves, we give an explicitation for an algebra of Hecke operators acting on E-cohomology. This leads to a vanishing result for Rezk's logarithmic cohomology operation on the units of E. It identifies a family of elements in the kernel with meromorphic modular forms whose Serre derivative is zero. Our calculation finds a connection to logarithms of modular units. In particular, we work out an action of Hecke operators on certain "logarithmic" q-series, in the sense of Knopp and Mason, that agrees with our vanishing result and extends the classical Hecke action on modular forms.

Explicit reciprocity laws for higher local fields

Using previously constructed reciprocity laws for the generalized Kummer pairing of an arbitrary (one-dimensional) formal group, in this article a special consideration is given to Lubin-Tate formal groups. In particular, this allows for a completely explicit description of the Kummer pairing in terms of multidimensional p-adic differentiation. The results obtained here constitute a generalization, to higher local fields, of the formulas of Artin-Hasse, Iwasawa, Kolyvagin and Wiles.

An equivariant Quillen theorem

A classical theorem due to Quillen (1969) identifies the unitary bordism ring with the Lazard ring, which represents the universal one-dimensional commutative formal group law. We prove an equivariant generalization of this result by identifying the homotopy theoretic Z/2-equivariant unitary bordism ring, introduced by tom Dieck (1970), with the Z/2-equivariant Lazard ring, introduced by Cole–Greenlees–Kriz (2000). Our proof combines a computation of the homotopy theoretic Z/2-equivariant unitary bordism ring due to Strickland (2001) with a detailed investigation of the Z/2-equivariant Lazard ring.

A Note on the Formal Groups of Weighted Delsarte Threefolds

One-dimensional formal groups over an algebraically closed field of positive characteristic are classified by their height. In the case of $K3$ surfaces, the height of their formal groups takes integer values between $1$ and $10$, or $\infty$. For Calabi-Yau threefolds, the height is bounded by $h^{1,2}+1$ if it is finite, where $h^{1,2}$ is a Hodge number. At present, there are only a limited number of concrete examples for explicit values or the distribution of the height. In this paper, we consider Calabi-Yau threefolds arising from weighted Delsarte threefolds in positive characteristic. We describe an algorithm for computing the height of their formal groups and carry out calculations with various Calabi-Yau threefolds of Delsarte type.

Formal groups and congruences

We give a criterion of integrality of a one-dimensional formal group law in terms of congruences satisfied by the coefficients of the canonical invariant differential. For an integral formal group law a p p -adic analytic formula for the local characteristic polynomial at p p is given. We demonstrate applications of our results to formal group laws attached to L L -functions, Artin–Mazur formal groups of algebraic varieties and hypergeometric formal group laws. This paper was written with the intention to give an explicit and self-contained introduction to the arithmetic of formal group laws, which would be suitable for non-experts. For this reason we consider only one-dimensional laws, though a generalization of our approach to higher dimensions is clearly possible. The ideas of congruences and p p -adic continuity in the context of formal groups were considered by many authors. We sketch the relation of our results to the existing literature in a separate paragraph at the end of the introductory section.

GROUP ACTION ON THE DEFORMATIONS OF A FORMAL GROUP OVER THE RING OF WITT VECTORS

A recent result by the authors gives an explicit construction for a universal deformation of a formal group $\unicode[STIX]{x1D6F7}$ of finite height over a finite field $k$ . This provides in particular a parametrization of the set of deformations of $\unicode[STIX]{x1D6F7}$ over the ring ${\mathcal{O}}$ of Witt vectors over $k$ . Another parametrization of the same set can be obtained through the Dieudonné theory. We find an explicit relation between these parameterizations. As a consequence, we obtain an explicit expression for the action of $\text{Aut}_{k}(\unicode[STIX]{x1D6F7})$ on the set of ${\mathcal{O}}$ -deformations of $\unicode[STIX]{x1D6F7}$ in the coordinate system defined by the universal deformation. This generalizes a formula of Gross and Hopkins and the authors’ result for one-dimensional formal groups.

A higher chromatic analogue of the image of J

We prove a higher chromatic analogue of Snaith's theorem which identifies the K-theory spectrum as the localisation of the suspension spectrum of CP^\infty away from the Bott class; in this result, higher Eilenberg-MacLane spaces play the role of CP^\infty = K(Z,2). Using this, we obtain a partial computation of the part of the Picard-graded homotopy of the K(n)-local sphere indexed by powers of a spectrum which for large primes is a shift of the Gross-Hopkins dual of the sphere. Our main technical tool is a K(n)-local notion generalising complex orientation to higher Eilenberg-MacLane spaces. As for complex-oriented theories, such an orientation produces a one-dimensional formal group law as an invariant of the cohomology theory. As an application, we prove a theorem that gives evidence for the chromatic redshift conjecture.

Arithmetic of π0-critical module

In this paper, for a specific kind of one-dimensional formal groups over the ring of integers of a local field in the case of small ramification we study the arithmetic of the formal module constructed on the maximal ideal of a local field, containing all the roots of the isogeny. This kind of formal groups is a little broader than Honda groups. The Shafarevich system of generators is constructed.

Formula omitted]-Coordinated modules for vertex algebras

We study ϕϵ-coordinated modules for vertex algebras, where ϕϵ with ϵ an integer parameter is a family of associates of the one-dimensional additive formal group. As the main results, we obtain a Jacobi type identity and a commutator formula for ϕϵ-coordinated modules. We then use these results to study ϕϵ-coordinated modules for vertex algebras associated to Novikov algebras by Primc.

On one-dimensional formal group laws in characteristic zero

Let \({\mathbb{K}}\) be a field of characteristic zero or, more generally, a \({\mathbb{Q}}\)-algebra. A formal power series \({F(x,y)=x+y+ \sum_{i,j \geq 1} a_{i,j}x^iy^j \in \mathbb{K}{[\![} x, y{]\!]}}\) is called a one-dimensional formal group law if F(F(x, y), z) = F(x, F(y, z)). Using some elementary methods, we prove that for every one-dimensional formal group law F(x, y) there exists a formal power series \({f(x)=x+\sum_{n\geq 2}f_nx ^n \in \mathbb{K}{[\![} x, y{]\!]}}\) so that F(x, y) = f−1(f(x) + f(y)).

Formal Hecke algebras and algebraic oriented cohomology theories

In the present paper, we generalize the construction of the nil Hecke ring of Kostant–Kumar to the context of an arbitrary formal group law, in particular, to an arbitrary algebraic oriented cohomology theory of Levine–Morel and Panin–Smirnov (e.g., to Chow groups, Grothendieck’s \(K_0\), connective \(K\)-theory, elliptic cohomology, and algebraic cobordism). The resulting object, which we call a formal (affine) Demazure algebra, is parameterized by a one-dimensional commutative formal group law and has the following important property: specialization to the additive and multiplicative periodic formal group laws yields completions of the nil Hecke and the 0-Hecke rings, respectively. We also introduce a formal (affine) Hecke algebra. We show that the specialization of the formal (affine) Hecke algebra to the additive and multiplicative periodic formal group laws gives completions of the degenerate (affine) Hecke algebra and the usual (affine) Hecke algebra, respectively. We show that all formal affine Demazure algebras (and all formal affine Hecke algebras) become isomorphic over certain coefficient rings, proving an analogue of a result of Lusztig.

Reciprocity laws through formal groups

A relation between formal groups and reciprocity laws is studied following the approach initiated by Honda. Let ξ \xi denote an m m th primitive root of unity. For a character χ \chi of order m m , we define two one-dimensional formal groups over Z [ ξ ] \mathbb {Z}[\xi ] and prove the existence of an integral homomorphism between them with linear coefficient equal to the Gauss sum of χ \chi . This allows us to deduce a reciprocity formula for the m m th residue symbol which, in particular, implies the cubic reciprocity law.

$${\phi}$$ -Coordinated Quasi-Modules for Quantum Vertex Algebras

We develop a theory of \({\phi}\) -coordinated (quasi-) modules for a general nonlocal vertex algebra where \({\phi}\) is what we call an associate of the one-dimensional additive formal group. By specializing \({\phi}\) to a particular associate, we obtain a new construction of weak quantum vertex algebras in the sense of Li (Selecta Mathematica (New Series) 11:349–397, 2005). As an application, we associate weak quantum vertex algebras to quantum affine algebras, and we also associate quantum vertex algebras and \({\phi}\) -coordinated modules to a certain quantum βγ-system explicitly.

Vertex [formula omitted]-algebras and their [formula omitted]-coordinated modules

In this paper, for every one-dimensional formal group F we formulate and study a notion of vertex F -algebra and a notion of ϕ -coordinated module for a vertex F -algebra where ϕ is what we call an associate of F . In the case that F is the additive formal group, vertex F -algebras are exactly ordinary vertex algebras. We give a canonical isomorphism between the category of vertex F -algebras and the category of ordinary vertex algebras. Meanwhile, for every formal group we completely determine its associates. We also study ϕ -coordinated modules for a general vertex Z -graded algebra V with ϕ specialized to a particular associate of the additive formal group and we give a canonical connection between V -modules and ϕ -coordinate modules for a vertex algebra obtained from V by Zhu’s change-of-variables theorem.

Application de Hodge-Tate duale d'un groupe de Lubin-Tate, immeuble de Bruhat-Tits du groupe linéaire et filtrations de ramification

One of the goals of this article is to describe the isomorphism between Lubin-Tate and Drinfeld towers at the level of their skeletons after taking quotient by GLn(OF)×OD× or I×OD×, where OD is the maximal order in the division algebra with invariant 1/n over F and I an Iwahori subgroup of GLn. We give applications to the theory of canonical subgroups on Lubin-Tate spaces, the description of spherical Hecke orbits in those spaces, fundamental domains for Hecke correspondences, and the Gross-Hopkins period mapping. We also study in detail the ramification filtrations (upper and lower) and the Hodge-Tate map of a one-dimensional formal p-divisible group

Arithmetic of the module of roots of the isogeny of a formal group in the case of small ramification

In this paper, for a one-dimensional formal group over the ring of integers of a local field in the case of small ramification we study the arithmetic of the module of roots of the isogeny, as well as the arithmetic of the formal module constructed on the maximal ideal of a local field containing all the roots of the isogeny. Bibliography: 5 titles.