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An approach to the construction of a basis of the Lazard ring by using the coefficients of a universal isogeny is described and examples of the application of this approach are given.

We introduce for each quiver $Q$ and each algebraic oriented cohomology theory $A$, the cohomological Hall algebra (CoHA) of $Q$, as the $A$-homology of the moduli of representations of the preprojective algebra of $Q$. This generalizes the $K$-theoretic Hall algebra of commuting varieties defined by Schiffmann-Vasserot. When $A$ is the Morava $K$-theory, we show evidence that this algebra is a candidate for Lusztig's reformulated conjecture on modular representations of algebraic groups. We construct an action of the preprojective CoHA on the $A$-homology of Nakajima quiver varieties. We compare this with the action of the Borel subalgebra of Yangian when $A$ is the intersection theory. We also give a shuffle algebra description of this CoHA in terms of the underlying formal group law of $A$. As applications, we obtain a shuffle description of the Yangian.

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.

Part of these notes was written as the author's 2013 master thesis. For proper flat schemes over a complete discrete valuation ring of mixed characteristic, we construct an isomorphism of certain subgroups of the Picard group and the first cohomology group of the structure sheaf. When the Picard scheme is available and smooth, it recovers the isomorphism coming from its formal completion. A reinterpretation of an old theorem of Mattuck is given., v2: exposition improved. 14 pages

We present a mathematical construction of new quantum information measures that generalize the notion of logarithmic negativity. Our approach is based on formal group theory. We shall prove that this family of generalized negativity functions, due their algebraic properties, is suitable for studying entanglement in many-body systems. Under mild hypotheses, the new measures are computable entanglement monotones. Also, they are composable: their evaluation over tensor products can be entirely computed in terms of the evaluations over each factor, by means of a specific group law. In principle, they might be useful to study separability and (in a future perspective) criticality of mixed states, complementing the role of R\'enyi's entanglement entropy in the discrimination of conformal sectors for pure states., Comment: 18 pages; to appear in Quantum Information Processing, 2021

In this work, we study generalized entropies and information geometry in a group-theoretical framework. We explore the conditions that ensure the existence of some natural properties and at the same time of a group-theoretical structure for a large class of entropies. In addition, a method for defining new entropies, using previously known ones with some desired group-theoretical properties is proposed. In the second part of this work, the information geometrical counterpart of the previous construction is examined and a general class of divergences are proposed and studied. Finally, a method of constructing new divergences from known ones is discussed; in particular, some results concerning the Riemannian structure associated with the class of divergences under investigation are formulated. © 2019 The Author(s) Published by the Royal Society. All rights reserved., M.A.R. has been partly supported by the research project no. FIS2015-63966, MINECO, Spain. P.T. has been partly supported by the research project FIS2015-63966, MINECO, Spain, by the ICMAT Severo Ochoa grant no. SEV-2015-0554 and by the GNFM, Italy.

This paper considers the problem of the description of unramified extensions of a local field which, together with the main field, do not contain nontrivial roots of isogeny of the corresponding formal group defined over a ring of integers of this field. This problem originated from investigation of extensions without higher ramification for multiplicative formal groups in the paper by Z.I. Borevich (1962).

This paper considers hyperbolic formal groups, which come from the elliptic curve theory, in the context of the theory of formal modules. In the first part of the paper, the characteristics of hyperbolic formal groups are considered, i.e., the explicit formulas for the formal logarithm and exponent; their convergence is studied. In the second part, the isogeny and its kernel and height are found; a p-typical logarithm is defined. The Artin–Hasse and Vostokov functions are then constructed; their correctness and main properties are evaluated.

Fontaine described the category of formal groups over the ring of Witt vectors over a finite field of characteristic p with the aid of triples consisting of the module of logarithms, the Dieudonné module, and the morphism from the former to the latter. We propose an explicit construction for the kernels in this category in term of Fontaine's triples. The construction is applied to the formal norm homomorphism in the case of an unramified extension of ℚp and of a totally ramiûed extension of degree less or equal than p. A similar consideration applied to a global extension allows us to establish the existence of a strict isomorphism between the formal norm torus and a formal group law coming from L-series.

In the present paper we extend the construction of the formal (affine) Demazure algebra due to Hoffnung, Malag\'on-L\'opez, Savage and Zainoulline in two directions. First, we introduce and study the notion of an extendable weight lattice in the Kac-Moody setting and show that all the definitions and properties of the formal (affine) Demazure operators and algebras hold for such lattices. Second, we show that for the hyperbolic formal group law the formal Demazure algebra is isomorphic (after extending the coefficients) to the Hecke algebra., Comment: Final version. Accepted for publication in Algebras and Representation Theory. ALGE-D-15-00163

The structure of the BP-cohomology of elementary abelian p-groups is studied, obtaining a presentation expressed in terms of BP-cohomology and mod-p singular cohomology, using the Milnor derivations. The arguments are based on a result on multi-Koszul complexes which is related to Margolis's criterion for freeness of a graded module over an exterior algebra.

The paper is devoted to problems at the intersection of formal group theory, the theory of Hirzebruch genera, and the theory of elliptic functions. In the focus of our interest are Tate formal groups corresponding to the general five-parametric model of the elliptic curve as well as formal groups corresponding to the general four-parametric Krichever genus. We describe coefficient rings of formal groups whose exponentials are determined by elliptic functions of levels 2 and 3.

We prove that iterative derivations on projective line cannot be expanded to iterative Hasse–Schmidt derivations, in the case when the iterativity rule is given by a non-algebraic formal group.

We provide several ingredients towards a generalization of the Littlewood-Richardson rule from Chow groups to algebraic cobordism. In particular, we prove a simple product-formula for multiplying classes of smooth Schubert varieties with any Bott-Samelson class in algebraic cobordism of the grassmannian. We also establish some results for generalized Schubert polynomials for hyperbolic formal group laws., Comment: 21 pages

Lie–Butcher (LB) series are formal power series expressed in terms of trees and forests. On the geometric side LB-series generalizes classical B-series from Euclidean spaces to Lie groups and homogeneous manifolds. On the algebraic side, B-series are based on pre-Lie algebras and the Butcher-Connes-Kreimer Hopf algebra. The LB-series are instead based on post-Lie algebras and their enveloping algebras. Over the last decade the algebraic theory of LB-series has matured. The purpose of this paper is twofold. First, we aim at presenting the algebraic structures underlying LB series in a concise and self contained manner. Secondly, we review a number of algebraic operations on LB-series found in the literature, and reformulate these as recursive formulae. This is part of an ongoing effort to create an extensive software library for computations in LB-series and B-series in the programming language Haskell.

We describe the coefficient rings of universal formal group laws which arise in algebraic geometry, algebraic topology and their application to mathematical physics. We also describe the homomorphisms of these coefficient rings coming from reductions of one formal group law to another. The proofs are based on the number-theoretic properties of binomial coefficients. Bibliography: 37 titles.

In this paper we present the results on the problem of explicit reciprocity law, as in the classical part of the field of algebraic numbers and their generalization to multidimensional fields and formal groups. The result is an explicit formula in the circular extension of the field of p-adic numbers is applied to the determination of a new public-key cryptosystems and electronic signatures.

We review some aspects of the theory of Lie algebras of (twisted and untwisted) formal pseudodifferential operators in one and several variables in a general algebraic context. We focus mainly on the construction and classification of nontrivial central extensions. As applications, we construct hierarchies of centrally extended Lie algebras of formal differential operators in one and several variables, Manin triples and hierarchies of nonlinear equations in Lax and zero curvature form.

We give a formula relating the transfer maps for the cohomology theories E n E_{n} and C t C_t to the transchromatic generalized character maps of a previous paper by the author. We then apply this to understand the effect of the transchromatic generalized character maps on Strickland’s isomorphism between the Morava E E -theory of the symmetric group Σ p k \Sigma _{p^k} (modulo a transfer ideal) and the global sections of the scheme that classifies subgroups of order p k p^k in the formal group associated to E n E_{n} . This provides an algebro-geometric interpretation to the C t C_t -cohomology of the class of groups arising as centralizers of finite sets of commuting elements in symmetric groups.

We give a formulation for descent of level structures on deformations of formal groups and study the compatibility between descent and a norm construction. Under this framework, we generalize Ando's construction of H ∞ complex orientations for Morava E-theories associated to the Honda formal groups over F p . We show the existence and uniqueness of such an orientation for any Morava E-theory associated to a formal group over an algebraic extension of F p and, in particular, orientations for a family of elliptic cohomology theories. These orientations correspond to coordinates on deformations of formal groups that are compatible with norm maps along descent.

Using the previously constructed explicit 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., 37 pages

In this note we define a generalization of Hall-Littlewood symmetric functions using formal group law and give an elementary proof of the generating function formula for the generalized Hall-Littlewood symmetric functions. We also give some applications of this formula., 10 pages, typo & minor corrections, Notation changed

In this survey we describe the construction of one-parameter subgroups (iteration groups) of Γ, the group of all (with respect to substitution) invertible power series in one indeterminate x over \(\mathbb{C}\). In other words, we describe all solutions of the translation equation in \(\mathbb{C}[\![\,x\,]\!]\), the ring of formal power series in x with complex coefficients. For doing this the method of formal functional equations will be applied. The coefficient functions of solutions of the translation equation are polynomials in additive and generalized exponential functions. Replacing these functions by indeterminates we obtain formal functional equations. Applying formal differentiation operators to these formal translation equations we obtain three types of formal differential equations. They can be solved in order to get explicit representations of the coefficient functions. For solving the formal differential equations we apply Briot–Bouquet differential equations in a systematic way.

We prove two theorems that confirm an observation of Lubin concerning families of $p$-adic power series that commute under composition: under certain conditions, there is a formal group such that the power series in the family are either endomorphisms of this group, or semi-conjugate to endomorphisms of this group., Comment: 9 pages. v3 : revised following referee's report

We calculate some formal group laws and genera closely related to the universal Buchstaber formal group law \(\mathcal{F}_B\).

In this paper, we investigate the representation of algebraic domains by means of Formal Concept Analysis. For a formal context, we can define a large number of consistent sets. Associated with each consistent set, there is a set of F-approximable concepts which are selected from the well known approximable concepts. By virtue of F-approximable concepts, formal contexts and algebraic domains are able to interpret each other. Moreover, by analyzing the finitely consistent sets, the algebraic bifinite domains, algebraic L-domains are exactly located at the corresponding formal contexts, respectively.

In the arithmetic of formal modules constructed on ideals of complete discrete valuation fields by using formal group laws, an essential role is played by the so-called primary elements, which give unramified extensions. In this paper, we obtain primary elements in an arbitrary formal module over a local field with perfect residue field. As formal group laws we consider so-called formal O0-module groups. In Section 1, we construct Hasse-type primary elements, whose definition involves an element of the integer ring of the completion of the maximal unramified extension of the field under consideration. In Section 2, we construct primary elements similar to those first appeared in [1, 2], whose definition involves only elements of the initial field. For this purpose, we expose Honda’s theory for formal O0modules and prove a theorem on the decomposition of the logarithm of a formal O0-module into a product of two factors on the basis of this theory. Then, we construct Artin–Hasse functions for universal formal O0-modules and, finally, prove the main theorems.

In 1969 Quillen discovered a deep connection between complex cobordism and formal group laws which he announced in [Qui69]. Algebraic topology has never been the same since. We will describe the content of [Qui69] and then discuss its impact on the field. This paper is a writeup of a talk on the same topic given at the Quillen Conference at MIT in October 2012. Slides for that talk are available on the author's home page.

A systematic description is given of properties of the group of formal power series in one variable with coefficients in a commutative unitary ring . This topological group has been studied intensively over the past 20 years, and a number of interesting results on its structure have been obtained. Here it is indicated how the group arises in several different areas of mathematics, such as complex cobordism or symplectic topology. Also considered is how the general structure of the group of complex formal power series is connected with classical problems of local uniformisation and the embedding of the germ of a holomorphic map in a flow. Bibliography: 115 titles.

We survey crystalline cohomology, crystals, and formal group laws with an emphasis on geometry. We apply these concepts to K3 surfaces, and especially to supersingular K3 surfaces. In particular, we discuss stratifications of the moduli space of polarized K3 surfaces in positive characteristic, Ogus’ crystalline Torelli theorem for supersingular K3 surfaces, the Tate conjecture, and the unirationality of K3 surfaces.

This paper surveys the current state of the theory of cobordism, focusing on geometric and universal properties of complex cobordism, the Landweber-Novikov algebra, and the formal group law of geometric cobordisms. The relationships with K-theory, algebraic cycles, formal group laws, compact Lie group actions on manifolds, toric topology, infinite-dimensional Lie algebras, and nilmanifolds are described. The survey contains key results and open problems. Bibliography: 124 titles.

A construction relating the theory of hyperfunctions with the theory of formal groups and generalizations of the classical Lipschitz summation formula is proposed. It involves new polylogarithmic rational functions constructed via the Fourier expansion of certain sequences of Bernoulli-type polynomials, related to the Lazard formal group. Related families of one-dimensional hyperfunctions are also constructed.

We provide a “geometric” proof of the reciprocity law for seventh powers, making use the arithmetic of the curve y = x+1/4, which is a hyperelliptic curve of genus 3. This proof is mainly the content of the Ph.D thesis of one of the authors([1]). We follow the ideas of D. Grant([2]), who gave a proof of a quintic reciprocity law based on the arithmetic of a hyperelliptic curve of genus 2. Our main tools are, like in D. Grant’s paper, the formal group in the origin of the Jacobian of the curve and the theorems of complex multiplication. However, there is a significant jump from the genus 2 to the genus 3 setting. For instance, in order to prove of complementary laws, we have to compute some 7-torsion points of the Jacobian; in our case, this computation become difficult due to the high dimension of the Riemann space involved. Along the proof of the complementary laws we produce some interesting units, similar to the classical elliptic units: they appear as the values on torsion points of some rational functions of the Jacobian.

The purpose of this paper is to introduce an algebraic cohomology and formal deformation theory of alternative algebras. A short review of basics on alternative algebras and their connections to some other algebraic structures is also provided.

We present a formalization of the topological ring of formal power series in Isabelle/HOL. We also formalize formal derivatives, division, radicals, composition and reverses. As an application, we show how formal elementary and hyper-geometric series yield elegant proofs for some combinatorial identities. We easily derive a basic theory of polynomials. Then, using a generic formalization of the fraction field of an integral domain, we obtain formal Laurent series and rational functions for free.

We determine the stable models of the modular curves X 0. p 3 / for primes p 13. An essential ingredient is the close relationship between the deformation theories of elliptic curves and formal groups, which was established in the Woods Hole notes of 1964. This enables us to apply results of Hopkins and Gross in our analysis of the supersingular locus.

The work is devoted to a wide class of formal groups, the ones given by polynomials, and to their relation to the Hilbert pairing. For the latter an explicit formula is obtained. The basic definitions are introduced in the work and the principal results are formulated, with the brief plans of proofs given for them. The detailed proofs are going to be given in the next work.

Let p > 3 be a prime. We consider j-zeros of Eisenstein series E k of weights k = p − 1 + M p a ( p 2 − 1 ) with M , a ⩾ 0 as elements of Q ¯ p . If M = 0 , the j-zeros of E p − 1 belong to Q p ( ζ p 2 − 1 ) by Hensel's lemma. Call these j-zeros p-adic liftings of supersingular j-invariants. We show that for every such lifting u there is a j-zero r of E k such that ord p ( r − u ) > a . Applications of this result are considered. The proof is based on the techniques of formal groups.

A formal Frobenius theorem, which is an analog of the classical integrability theorem for smooth distributions, is proved and applied to generalize the argument shift method of A. S. Mishchenko and A. T. Fomenko to finite-dimensional Lie algebras over any field of characteristic zero. A completeness criterion for a commutative set of polynomials constructed by the formal argument shift method is obtained.

We study the variation of the unit roots of members of the Dwork families of Calabi–Yau varieties over a finite field by the method of Dwork–Katz and also from the point of view of formal group laws. A p-adic analytic formula for the unit roots away from the Hasse locus is obtained.

Lubin conjectures that for an invertible series to commute with a noninvertible series with only simple roots of iterates, two such commuting power series must be endomorphisms of a single formal group. In this paper, we show that if the reduction of these two commuting power series are endomorphisms of a formal group, then themselves are endomorphisms of a formal group.

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.