Formal Group Theory Research Articles

New computable entanglement monotones from formal group theory

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 Rényi’s entanglement entropy in the discrimination of conformal sectors for pure states.

Multivariate group entropies, super-exponentially growing complex systems, and functional equations.

We define the class of multivariate group entropies as a novel set of information-theoretical measures, which extends significantly the family of group entropies. We propose new examples related to the "super-exponential" universality class of complex systems; in particular, we introduce a general entropy, representing a suitable information measure for this class. We also show that the group-theoretical structure associated with our multivariate entropies can be used to define a large family of exactly solvable discrete dynamical models. The natural mathematical framework allowing us to formulate this correspondence is offered by the theory of formal groups and rings.

A new class of entropic information measures, formal group theory and information geometry.

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.

Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory.

The entropy of Boltzmann-Gibbs, as proved by Shannon and Khinchin, is based on four axioms, where the fourth one concerns additivity. The group theoretic entropies make use of formal group theory to replace this axiom with a more general composability axiom. As has been pointed out before, generalised entropies crucially depend on the number of allowed degrees of freedom N. The functional form of group entropies is restricted (though not uniquely determined) by assuming extensivity on the equal probability ensemble, which leads to classes of functionals corresponding to sub-exponential, exponential or super-exponential dependence of the phase space volume W on N. We review the ensuing entropies, discuss the composability axiom and explain why group entropies may be particularly relevant from an information-theoretical perspective.

Двойственность в абелевых многообразиях и формальных группах над локальными полями

Статья посвящена памяти Олега Николаевича Введенского (1937 – 1981 гг.). О. Н. Введенский был учеником академика И. Р. Шафаревича. Исследования О. Н. и полученные им результаты связаны с двойственностью в эллиптических кривых и с соответствующими когомологиями Галуа над локальными полями, со спариванием Шафаревича-Тэйта и с другими спариваниями, с локальной и квази-локальной теорией полей классов эллиптических кривых, с теорией абелевых многообразий размерности больше 1, с теорией коммутативных формальных групп над локальными полями. Представлены как результаты, полученные О. Н. Введенским, так и новые избранные результаты, развивающие исследования в направлениях фундаментальных групп схем, главных однородных пространств (торсеров) и двойственности. Первая часть статьи, представлення здесь, является введением как в результаты, полученные О. Н. Введенским в направлении двойственности абелевых многообразий и формальных групп, так и в новые избранные результаты, развивающие исследования в направлениях фундаментальных групп схем, главных однородных пространств (торсеров) и двойственности. Во Введении приведены предварительные сведения и представлено содержание статьи. В первом разделе дан краткий обзор избранных результатов по теории алгебраических, квазиалгебраические и проалгебраические группы и групповых схем. Далее, в разделе 2 преставлены избранные результаты по фундаментальным группам алгебраических многообразий, по фундаментальным группам схем, а в разделе 3 - избранные результаты о главных однородных пространствах (торсерах), развивающие исследования О. Н. и других авторов. Термин торсер мы используем как перевод на русский язык в редакции И.Р. Шафаревича английского термина torsor. В разделе 4 даны сведения о двойственности, а в разделе 5 представлены результаты О. Н. по арифметической теории формальных групп и их развитие. Результаты, этого раздела, представленные над локальными и квази-локальными полями K, над их кольцами целых, и над их полями вычетов k, связанны (1) с формальной структурой абелевых многообразий, (2) с коммутативными формальными группами, (3) с соответствующими гомоморфизмами и изогениями. В статье алгебраические многообразия, абелевы схемы и коммутативные формальные групповые схемы определены, как правило, над локальными и квази-локальными полями, над их кольцами целых, и над их полями вычетов. Но кратко рассматриваются эти объектыи и над глобальными полями, так как О. Н. интересовала тематика алгебраических многообразий над глобальными полями и он проводил соответствующие исследования. Предполагается, что характеристика полей вычетов больше 3, если не оговаривается иное.Я признателен В.Н. Чубарикову за предложение опубликовать статью в сборнике.Особая признательность Н. М. Добровольскому за помощь и поддержку в процессе подготовки статьи к печати.

Formal groups and Z-entropies.

We shall prove that the celebrated Rényi entropy is the first example of a new family of infinitely many multi-parametric entropies. We shall call them the Z-entropies. Each of them, under suitable hypotheses, generalizes the celebrated entropies of Boltzmann and Rényi. A crucial aspect is that every Z-entropy is composable (Tempesta 2016 Ann. Phys.365, 180-197. (doi:10.1016/j.aop.2015.08.013)). This property means that the entropy of a system which is composed of two or more independent systems depends, in all the associated probability space, on the choice of the two systems only. Further properties are also required to describe the composition process in terms of a group law. The composability axiom, introduced as a generalization of the fourth Shannon-Khinchin axiom (postulating additivity), is a highly non-trivial requirement. Indeed, in the trace-form class, the Boltzmann entropy and Tsallis entropy are the only known composable cases. However, in the non-trace form class, the Z-entropies arise as new entropic functions possessing the mathematical properties necessary for information-theoretical applications, in both classical and quantum contexts. From a mathematical point of view, composability is intimately related to formal group theory of algebraic topology. The underlying group-theoretical structure determines crucially the statistical properties of the corresponding entropies.

Supercongruences and complex multiplication

We study congruences involving truncated hypergeometric series of the formF23(1/2,1/2,1/21,1;λ)(mps−1)/2:=∑k=0(mps−1)/2((1/2)k/k!)3λk where p is a prime and m,s are positive integers. These truncated hypergeometric series are related to the arithmetic of a family of K3 surfaces. For special values of λ, with s=1, our congruences are stronger than those predicted by the theory of formal groups, because of the presence of elliptic curves with complex multiplications. They generalize a conjecture made by Stienstra and Beukers for the λ=1 case and confirm some other supercongruence conjectures at special values of λ.

Coefficient rings of Tate formal groups determining Krichever genera

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.

The Lazard formal group, universal congruences and special values of zeta functions

A connection between the theory of formal groups and arithmetic number theory is established. In particular, it is shown how to construct general Almkvist–Meurman–type congruences for the universal Bernoulli polynomials that are related with the Lazard universal formal group (based on earlier works of the author). Their role in the theory of L L –genera for multiplicative sequences is illustrated. As an application, sequences of integer numbers are constructed. New congruences are also obtained, useful to compute special values of a new class of Riemann–Hurwitz–type zeta functions.

Using Atomic Orbitals and Kinesthetic Learning To Authentically Derive Molecular Stretching Vibrations

The stretching modes of MLx complexes have the same symmetry as the atomic orbitals on M that are used to form its σ bonds. In the exercise suggested here, the atomic orbitals are used to derive the form of the stretching modes without the need for formal group theory. The analogy allows students to help understand many conceptually difficult aspects of the vibrational spectroscopy of polyatomic molecules, including IR activity, degeneracy, and orthogonality. The approach intentionally requires active participation and is nonmathematical and “low tech”. It is suitable for large classes of chemistry majors and nonmajors.

Hyperfunctions, formal groups and generalized Lipschitz summation formulas

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.

Group entropies, correlation laws, and zeta functions

The notion of group entropy is proposed. It enables the unification and generaliztion of many different definitions of entropy known in the literature, such as those of Boltzmann-Gibbs, Tsallis, Abe, and Kaniadakis. Other entropic functionals are introduced, related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium when the dynamics is weakly chaotic. The associated thermostatistics are discussed. The mathematical structure underlying our construction is that of formal group theory, which provides the general structure of the correlations among particles and dictates the associated entropic functionals. As an example of application, the role of group entropies in information theory is illustrated and generalizations of the Kullback-Leibler divergence are proposed. A new connection between statistical mechanics and zeta functions is established. In particular, Tsallis entropy is related to the classical Riemann zeta function.

The canonical subgroup: a "subgroup-free" approach

Beyond the crucial role they play in the foundations of the theory of overconvergent modular forms, canonical subgroups have found new applications to analytic continuation of overconvergent modular forms. For such applications, it is essential to understand various "numerical" aspects of the canonical subgroup, and in particular, the precise extent of its overconvergence. In this paper, we develop a theory of canonical subgroups for a general class of curves (including the unitary and quaternionic Shimura curves), using formal and rigid geometry. In our approach, we use the common geometric features of these curves rather than their (possible) specific moduli-theoretic description; it allows us to reproduce, for the classical cases, the optimal radii of definition for the canonical subgroup, usually derived by employing the theory of formal groups.

Moduli space of filtered λ‐ringstructures over a filtered ring

Motivated in part by recent works on the genus of classifying spaces of compact Lie groups, here we study the set of filtered λ‐ring structures over a filtered ring from a purely algebraic point of view. From a global perspective, we first show that this set has a canonical topology compatible with the filtration on the given filtered ring. For power series rings R[[x]], where R is between ℤ and ℚ, with the x‐adic filtration, we mimic the construction of the Lazard ring in formal group theory and show that the set of filtered λ‐ring structures over R[[x]] is canonically isomorphic to the set of ring maps from some “universal” ring U to R. From a local perspective, we demonstrate the existence of uncountably many mutually nonisomorphic filtered λ‐ring structures over some filtered rings, including rings of dual numbers over binomial domains, (truncated) polynomial, and power series rings over ℚ‐algebras.

Generalized group characters and complex oriented cohomology theories

Let BG be the classifying space of a finite group G. Given a multiplicative cohomology theory E ⁄ , the assignment G 7i! E ⁄ (BG) is a functor from groups to rings, endowed with induction (transfer) maps. In this paper we investigate these functors for complex oriented cohomology theories E ⁄ , using the theory of complex representations of finite groups as a model for what one would like to know. An analogue of Artin's Theorem is proved for all complex oriented E ⁄ : the abelian subgroups of G serve as a detecting family for E ⁄ (BG), modulo torsion dividing the order of G. When E ⁄ is a complete local ring, with residue field of characteristic p and associated formal group of height n, we construct a character ring of class functions that computes 1 E ⁄ (BG). The domain of the characters is Gn,p, the set of n-tuples of elements in G each of which has order a power of p. A formula for induction is also found. The ideas we use are related to the Lubin Tate theory of formal groups. The construction applies to many cohomology theories of current interest: completed versions of elliptic cohomology, E ⁄ n- theory, etc. The nth Morava K-theory Euler characteristic for BG is computed to be the number of G-orbits in Gn.p. For various groups G, including all symmetric groups, we prove that K(n)⁄(BG) concentrated in even degrees. Our results about E⁄(BG) extend to theorems about E⁄(EG◊GX), where X is a finite G-CW complex.

Products on 𝑀𝑈-modules

Elmendorf, Kriz, Mandell and May have used their technology of modules over highly structured ring spectra to give new constructions of M U MU -modules such as B P BP , K ( n ) K(n) and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over M U [ 1 2 ] ∗ MU[\frac {1}{2}]_* that are concentrated in degrees divisible by 4 4 ; this guarantees that various obstruction groups are trivial. We extend these results to the cases where 2 = 0 2=0 or the homotopy groups are allowed to be nonzero in all even degrees; in this context the obstruction groups are nontrivial. We shall show that there are never any obstructions to associativity, and that the obstructions to commutativity are given by a certain power operation; this was inspired by parallel results of Mironov in Baas-Sullivan theory. We use formal group theory to derive various formulae for this power operation, and deduce a number of results about realising 2 2 -local M U ∗ MU_* -modules as M U MU -modules.

Symmetric Functions, Formal Group Laws, and Lazard's Theorem

Lazard's theorem is a central result in formal group theory; it states that the ring over which the universal formal group law is defined (known as the Lazard ring) is a polynomial algebra over the integers with infinitely many generators. This ring also shows up in algebraic topology as the complex cobordism ring. The main aim of this paper is to show that the polynomial structure of the Lazard ring follows from the polynomial structure of a certain subalgebra of symmetric functions with integer coefficients. The connection between symmetric functions and the Lazard ring is provided by a certain Hopf algebra map from symmetric functions to the covariant bialgebra of a formal group law. We study this map by deriving formulas for the images of certain symmetric functions; in passing, we use this map to prove some symmetric function and Catalan number identities. Based on the above results, we prove Lazard's theorem, and present an application to the construction of certainp-typical formal group laws over the integers. Combinatorial methods play a major role throughout this paper.

Hopf algebras of set systems

Hopf algebras play a major rôle in such diverse mathematical areas as algebraic topology, formal group theory, and theoretical physics, and they are achieving prominence in combinatorics through the influence of G.-C. Rota and his school. Our primary purpose in this article is to build on work of Schmitt [18,19], and establish combinatorial models for several of the Hopf algebras associated with umbral calculus and formal group laws. In so doing, we incorporate and extend certain invariants of simple graphs such as the umbral chromatic polynomial, and Stanley's [21] recently introduced symmetric function. Our fundamental combinatorial components are finite set systems, together with a versatile generalization in which they are equipped with a group of automorphisms. Interactions with the Roman-Rota umbral calculus over graded rings of scalars which may contain torsion are a significant feature of our presentation.

Hopf Galois Kummer theory of formal groups

Let ƒ : F → G be an isogeny between finite n -dimensional formal groups defined over R , the valuation ring of some field extension K of Q p . Let H be the R -Hopf algebra which arises from this isogeny. For such H , we classify Gal( H ), the group of H -Galois objects. Let M be the maximal ideal of R , and let P ( F, K ) denote the n -tuples of M under the group operation induced by F . Our main result is the construction of an isomorphism from the cokernel of P ( ƒ ) to Gal( H ), where P ( ƒ ) is the induced map from P ( F, K ) to P ( G, K ). In geometric language Gal( H ) describes the group of isomorphism classes of principal homogeneous spaces for Spec( H ) over Spec( R ). Geometric methods have been used by Mazur to establish the above isomorphism, but the proof is nonconstructive. A geometric approach has also provided a formula for the cardinality of Gal( H ). We give an alternative derivation of this result using formal group techniques.

The formal group of the Jacobian of an algebraic curve

In this paper we give an explicit construction of the formal group of the Jacobian of an algebraic curve using a basis for the holomorphic differentials on the curve at a rational non-Weierstrass point. We construct the formal group of the Jacobian of the modular curve Xo(l) and using a result of T. Honda, we prove that this formal group is p- integral for all but finitely many p . Introduction. Formal group laws have proven to be very useful tools in many areas of mathematics and computer science. In particular, the formal group of an elliptic curve has been used to great effect in elliptic curve theory (for details see for example Silverman (13)) and the use of the formal group of an abelian variety is pervasive in arithmetic and algebraic geometry (see Shatz (11) or Milne (8)). Despite the fact that explicit formulae have been useful in the elliptic curve case, explicit examples of the formal group of other abelian varieties are few. Recently, Grant (4) and Flynn (3) have independently given explicit constructions of the formal group of the Jacobian of curves of genus two. Grant uses classical formulae for genus two theta functions to give explicit defining equations for the Jacobian and a set of param- eters for its group law in a specific P8 embedding. His embedding requires that the curve have a Weierstrass point defined over the base field. Πynn's result does not assume the existence of a rational Weier- strass point and thus he must use a P 15 embedding of the Jacobian. This paper gives an explicit construction of the formal group of the Jacobian using a basis for the holomorphic differentials on the curve. The construction generalizes that of the formal group of an elliptic curve. In §1, we review some of the basic facts about higher dimensional formal groups. Hazewinkel (5) is a good reference for the theory of formal groups in general. Section II details the construction of the formal group of the Jacobian. Since this construction depends only on a basis for the holomorphic differentials on the curve at a non- Weierstrass point, it is especially useful in cases where much is known