Universal Formal Group Research Articles

Universal Formal Group for Elliptic Genus of Level N

An elliptic function of level N determines an elliptic genus of level N as a Hirzebruch genus. It is known that any elliptic function of level N is a specialization of the Krichever function that determines the Krichever genus. The Krichever function is the exponential of the universal Buchstaber formal group. In this work we give a specialization of the Buchstaber formal group such that this specialization determines formal groups corresponding to elliptic genera of level N. Namely, an elliptic function of level N is the exponential of a formal group of the form F(u, v) = (u2A(v) − v2A(u))/(uB(v) − vB(u)), where A(u), B(u) ∈ ℂ[[u]] are power series with complex coefficients such that A(0) = B(0) = 1, A″(0) = B′(0) = 0, and for m = [(N − 2)/2] and n = [(N − 1)/2] there exist parameters (a1, …, am, b1, …, bn) for which the relation \(\prod\nolimits_{j=1}^{n-1}\left(B(u)+b_{j} u\right)^{2} \cdot\left(B(u)+b_{n} u\right)^{N-2 n}=A(u)^{2} \prod\nolimits_{k=1}^{m-1}\left(A(u)+a_{k} u^{2}\right)^{2} \cdot\left(A(u)+a_{m} u^{2}\right)^{N-1-2 m}\) holds. For the universal formal group of this form, the exponential is an elliptic function of level at most N. This statement is a generalization to the case N > 2 of the well-known result that the elliptic function of level 2 determining the elliptic Ochanine–Witten genus is the exponential of a universal formal group of the form F(u, v) = (u2 − v2)/(uB(v) − vB(u)), where B(u) ∈ ℂ[[u]], B(0) = 1, and B′(0) = 0. We prove this statement for N = 3, 4, 5, 6. We also prove that the elliptic function of level 7 is the exponential of a formal group of this form. Universal formal groups that correspond to elliptic genera of levels N = 5, 6, 7 are obtained in this work for the first time.

Buchstaber formal group and elliptic functions of small levels

In the paper, we suggest a method for finding relations concerning series defining the Buchstaber formal group. This method is applied to the cases in which the exponent of the group is an elliptic function of level n = 2, 3, and 4. An algebraic relation for the series defining the universal Buchstaber formal group is also proved.

Elliptic function of level 4

The work is dedicated to the theory of elliptic functions of level $n$. An elliptic function of level $n$ determines a Hirzebruch genus that is called elliptic genus of level $n$. Elliptic functions of level $n$ are also interesting as solutions of Hirzebruch functional equations. The elliptic function of level $2$ is the Jacobi elliptic sine. It determines the famous Ochanine--Witten genus. It is the exponential of the universal formal group of the form \[ F(u,v)=\frac{u^2 -v^2}{u B(v) - v B(u)}, \quad B(0) = 1. \] The elliptic function of level $3$ is the exponential of the universal formal group of the form \[ F(u,v)=\frac{u^2 A(v) -v^2 A(u)}{u A(v)^2 - v A(u)^2}, \qquad A(0) = 1, \quad A"(0) = 0. \] In this work we have obtained that the elliptic function of level $4$ is the exponential of the universal formal group of the form \[ F(u,v)=\frac{u^2 A(v) -v^2 A(u)}{u B(v)-v B(u)}, \text{ where } A(0) = B(0) = 1, \] and for $B'(0) = A"(0) = 0, A'(0) = A_1, B"(0) = 2 B_2$ the relation holds \[ (2 B(u) + 3 A_1 u)^2 = 4 A(u)^3 - (3 A_1^2 - 8 B_2) u^2 A(u)^2. \] To prove this result we have expressed the elliptic function of level $4$ in terms of Weierstrass elliptic functions.

Coefficient rings of formal group laws

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.

Beyond the Shannon–Khinchin formulation: The composability axiom and the universal-group entropy

The notion of entropy is ubiquitous both in natural and social sciences. In the last two decades, a considerable effort has been devoted to the study of new entropic forms, which generalize the standard Boltzmann–Gibbs (BG) entropy and could be applicable in thermodynamics, quantum mechanics and information theory. In Khinchin (1957), by extending previous ideas of Shannon (1948) and Shannon and Weaver (1949), Khinchin proposed a characterization of the BG entropy, based on four requirements, nowadays known as the Shannon–Khinchin (SK) axioms.The purpose of this paper is twofold. First, we show that there exists an intrinsic group-theoretical structure behind the notion of entropy. It comes from the requirement of composability of an entropy with respect to the union of two statistically independent systems, that we propose in an axiomatic formulation. Second, we show that there exists a simple universal family of trace-form entropies. This class contains many well known examples of entropies and infinitely many new ones, a priori multi-parametric. Due to its specific relation with Lazard’s universal formal group of algebraic topology, the new general entropy introduced in this work will be called the universal-group entropy. A new example of multi-parametric entropy is explicitly constructed.

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.

On the Buchstaber formal group law and some related genera

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

Computing the Krichever genus

Let $\psi$ denote the genus that corresponds to the formal group law having invariant differential $\omega(t)$ equal to $\sqrt{1+p_1t+p_2t^2+p_3t^3+p_4t^4}$ and let $\kappa$ classify the formal group law strictly isomorphic to the universal formal group law under strict isomorphism $x\CP(x)$. We prove that on the rational complex bordism ring the Krichever-H\"ohn genus $\phi_{KH}$ is the composition $\psi\circ \kappa^{-1}$. We construct certain elements $A_{ij}$ in the Lazard ring and give an alternative definition of the universal Krichever formal group law. We conclude that the coefficient ring of the universal Krichever formal group law is the quotient of the Lazard ring by the ideal generated by all $A_{ij}$, $i,j\geq 3$.

L-series and Hurwitz zeta functions associated with the universal formal group

L-series and Hurwitz zeta functions associated with the universal formal group

On the classifying ring for Abel formal group laws

The ring over which the universal Abel formal group law is defined is characterized by an integrality condition. Three localizations of this ring considered by Bukhshtaber and Kholodov are given simple descriptions in terms of integer-valued polynomials, and the Abel formal group over one of these rings is shown to be multiplicative.

On the BP-homology of 2e × 2e

AbstractWe study υ0- and υ1-divisibility properties of the [2e]-series associated to the universal 2-typical formal group law. This allows us to identify elements annihilating the toral class τ in BP*(2e × 2e). We conjecture that these elements form a minimal system of generators of the annihilator ideal of τ. This would provide a Landweber-type presentation for the BP-homology of 2e × 2e from which the relation hom:dimBP (Z2e × Z2e) = 2 would be an easy consequence.

Arbeitsgemeinschaft mit aktuellem Thema: Algebraic Cobordism

Over the years, many different types and flavors of cohomology theories for algebraic varieties have been constructed. Theories like étale cohomology or de Rham cohomology provide algebraic versions of the topological theory of singular cohomology. The Chow ring and algebraic K_0 are other (partial) examples, more directly tied to algebraic geometry. The partial theory K_0^{alg} was extended to a full theory with the advent of Quillen's higher algebraic K -theory. It took considerably longer for the Chow ring to be extended to motivic cohomology. In the process of doing so, Voevodsky developed his category of motives, and this construction was put in a more general setting with the development by Morel–Voevodskyof of \mathbb{A}^1 homotopy theory. This enabled a systematic construction of cohomology theories on algebraic varieties, with algebraic K -theory and motivic cohomology being only two fundamental examples. These two cohomology theories have in common the existence of a good theory of push-forward maps for projective morphisms. Not all cohomology theories have this structure, those that do are called oriented . In the Morel–Voevodsky stable homotopy category, the universal oriented theory is represented by the \mathbb{P}^1 -spectrum M\mathbb G\ell , an algebraic version of the classical Thom spectrum MU . The corresponding cohomology theory M\mathbb G\ell^{*,*} is called higher algebraic cobordism . In an attempt to better understand the theory M\mathbb G\ell^{*,*} , Levine and Morel constructed a theory of algebraic cobordism \Omega^* . This is (conjecturally) related to M\mathbb G\ell^{*,*} as the classical Chow ring \mathrm{CH}^* is to motivic cohomology and like \mathrm{CH}^* , \Omega^* has a purely algebro-geometric description. In addition to giving some insight into M\mathbb G\ell^{*,*} , \Omega^* gives a simultaneous presentation of both \mathrm{CH}^* and K_0 , exhibiting K_0 as a deformation of \mathrm{CH}^* . \Omega^* has also been used to give conceptually simple proofs of various “degree formulas” first formulated by Rost. These degree formulas have been used in the study of Pfister quadrics and norm varieties, properties of which are used in the proofs of the Milnor conjecture and the Bloch–Kato conjecture. In this workshop, we describe aspects of the topological theory of complex cobordism which are important for algebraic cobordism (Lectures 1-3) and give the construction of \Omega^* and proofs of its fundamental properties (Lectures 4-7). In lectures 8-11, we show how K_0 and \mathrm{CH}^* are described by \Omega^* , how \Omega^* recovers the universal formal group law, give the proof the generalized degree formula for \Omega^* and use this to proof the degree formula for the Segre class. Additional applications to Steenrod operations, further degree formulas and the use of these in the study of quadrics and other varietes is given in lectures 12 and 13. Lectures 14 and 15 concern the construction of funtorial pull-backs in algebraic cobordism. The two concluding lectures (16 and 17) give a quick sketch of the Morel–Voevodsky \mathbb{A}^1 stable homotopy category and describe what we know about M\mathbb G\ell and its relation to motivic cohomology and \Omega^* . The workshop Algebraic Cobordism ,organised by Marc Levin (Boston) and Fabien Morel (München) was held April 4th–April 8th, 2005. This meeting was well attended with 55 participants.

P-adic period map for the moduli space of deformations of a formal group

The moduli space of deformations of a formal group over a finite field is studied. We consider Lubin–Tate and Dieudonné approaches and find an explicit relation between them employing Hazewinkel's universal p-typical formal group, Honda's theory and rigid power series. The formula obtained allows to give an explicit description of the action of the automorphism group of the formal group on the moduli space. It essentially generalizes an analogous result of Gross and Hopkins [Contemp. Math. 158 (1994) 23–88].

Modified Stirling Numbers and p -Divisibility in the Universal Typical p k -Series

The 1-dimensional universal formal group law is a power series (in two variables and with coefficients in Lazard's ring) carrying a lot of geometrical and algebraic properties. For a prime p, we study the corresponding “p-localized” formal group law through its associated p k-series, [p k](x) = Σ s≥0 a k,s x s(p−1)+1—the p k-fold iterated formal sum of a variable x. The coefficients a k,s lie in the Brown-Peterson ring BP * = ℝ(p)[v 1,v 2,...] and we describe part of their structure as polynomials in the variables v i with p-local coefficients. This is achieved by introducing a family of filtrations {W ϕ}ϕ≥1 in BP * and studying the value of a k,s in each of the associated (bi)graded rings BP */W ϕ. This allows us to identify, among monomials in a k,s of minimal W ϕ-filtration (1 ≤ ϕ ≤ k), an explicit monomial m ϕ,k,s carrying the lowest possible p-divisibility. The p-local coefficient of m ϕ,k,s is described as a Stirling-type number of the second kind and its actual value is computed up to p-local units. It turns out that m k,k,s not only carries the lowest W k -filtration but, more importantly, the lowest p-divisibility among all other monomials in a k,s . In particular, we obtain a complete description of the p-divisibility properties of each a k,s .

Realization of Abel's universal formal group law

To a complex oriented cohomology theory $h^*(-)$ one may assign a formal group law $F_h(u,v)$ over $h^*(pt)$ . The purpose of this paper is to show that the converse holds true for the case of Abel's universal formal group law ${\cal F}_{Ab}(u,v)$ , i.e. we will prove the existence of a complex oriented cohomology theory $Ab^*(-)$ whose formal group law $F_{Ab}$ is isomorphic to ${\cal F}_{Ab}(u,v)$ .

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.

Roots of Unity and the Adams-Novikov Spectral Sequence for Formal A-Modules

The cohomology of a Hopf algebroid related to the Adams-Novikov spectral sequence for formal A-modules is studied in the special case in which A is the ring of integers in the field obtained by adjoining pth roots of unity to ?2p, the p-adic numbers. Information about these cohomology groups is used to give new proofs of results about the E2 term of the Adams spectral sequence based on 2-local complex K-theory, and about the odd primary Kervaire invariant elements in the usual Adams-Novikov spectral sequence. One of the most powerful tools used in the computation of stable homotopy groups is the Adams-Novikov spectral sequence. The E2 term of this spectral sequence is a certain Ext group derived from a universal formal group law. In [R3] the corresponding Ext group for a universal formal A-module, for A the ring of algebraic integers in an algebraic number field, K, or its padic completion, was introduced and certain conjectures about these groups were formulated. One of these conjectures (concerning the value of ExtlI* ) was confirmed in [J] using a Hopf algebroid (i.e., a generalized Hopf algebra in which the left and right units need not agree), EAT, which generalizes the Hopf algebroid K*K of stable cooperations for complex K-theory. The present paper is concerned with the cohomology of EAT in the special case of A = p where C is a pth root of unity and Zp denotes the p-adic integers. We will show that in this case EAT is contained in an extension of Hopf algebroids EA )EA T EAT and that the cohomology of EAT can be completely described. This provides us with information about the cohomology of EAT via the Cartan-Eilenberg spectral sequence associated to this extension. Two applications of this result are presented. In the case p = 2, EAT can be identified with the 2-adic completion of the Hopf algebroid K*K(2) of stable cooperations for 2-primary complex K-theory. In this case the cohomology Received by the editors February 1, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 55T25; Secondary 55N22, 144L05. ? 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page

On Drinfeld's universal formal group over thep-adic upper half plane

All of our results are stated for 2-dimensional modules with action by the quaternion division algebra overQ p . Drinfeld's results are true in much greater generality. We remark that our results generalize easily to the case of 2-dimensional modules with action by quaternion algebras over extensions ofQ p by applying the theory of formal $$\mathcal{O}$$ -modules. We suspect that Drinfeld's higher dimensional modules over $$W(\bar F_p )$$ are determined by formulae similar to that in Theorem 46, but with α and β generalized to moduli for higher dimensionalQ p -subspaces of $$\hat Q_{p}^{ur} $$ ; however, we have not investigated this in any detail. Although this work amplifies Drinfeld's original paper by supplying many details in certain cases, it is seriously limited in that it considers lifts of SFD modules to unramified rings only. The most interesting points in thep-adic upper half plane are the points defined over ramified rings, which reduce modp to the singular points on the special fiber. What happens there? We do not have a simple answer. Drinfeld's moduli for formal groups on thep-adic upper half plane is the basis for his proof that Shimura curves havep-adic uniformizations. In a later work, we hope to exploit improved versions of the techniques in this work to better understand the arithmetic of Shimura curves. In particular, in the course of work onp-adicL-functions, we have been led to construct certain “p-adic periods” associated to the cohomology of sheaves on Shimura curves which depend essentially on the existence of ap-adic uniformization. We hope to use Drinfeld's moduli to obtain a more natural construction of these periods in terms of the Gauss-Manin connection, and thereby to gain a better understanding of how they might come to appear in special values ofp-adicL-functions.

Actions on Invariant Spheres around Isolated Fixed Points of Actions of Cyclic Groups

Fix a prime number p and let Zp be a cyclic group of order p. We consider a pair (M, 0) consisting of a compact simply connected almost complex manifold M without boundary and a smooth Zp-action 0: ZPX M—>M preserving the almost complex structure of M. We suppose that M is given an invariant Riemannian metric. If a(eM) is an isolated fixed point, then the induced action of Zp on the tangent space at a gives a complex Zp-module Va which has no trivial irreducible factor. Let f : EZp—*BZp be a universal principal Zp-bundle and let f (Vtt) : EZpXzp Va —>BZP be the V0-bundle associated with ?. If a and b are isolated fixed points, we compare the cobordism Euler classes 0(f (Va)) and e(£ (Vb)) which belong to the complex cobordism group MU* (BZP) of the classifying space BZP of Zp. Let .PV be the universal formal group law over MU*9 and write

Remarks on the Coefficient Ring of Quaternionic Oriented Cohomology Theories

Let h* be a complex (or real) oriented cohomology theory. It has recently been observed that the of h* plays a very important role in such a theory. Above all, D. Quillen [6] showed that the formal group FU of the complex cobordism theory MU* ( ) is isomorphic to the Lazard universal formal group, and its coefficients generate the ground ring MU* (pt). Unfortunately, in the case of quaternionic oriented cohomology theory (see § 2), there exists no such formal group, since the tensor product of quaternionic line bundles does not yield a quaternionic bundle. This situation makes it difficult, for example, to produce enough generators of MSp*(pt). However there are some substitutes for the formal group. In particular, using the total Pontrjagin class of a certain quaternionic vector bundle 2C (see § 3), N. Ja. Gozman [5] defined a subring A of MU* (pt) which is contained in the image of the forgetful homomorphism (p: MSp* (pt}-*MU*(pt). In this paper, we will generalize his approach to arbitrary quaternionic oriented cohomology theory 7i*, and define a subring Ah of h* (pt) which is generated by the coefficients of certain power series. Then it will be shown that