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Since the development of higher local class field theory, several explicit reciprocity laws have been constructed. In particular, there are formulas describing the higher-dimensional Hilbert symbol given, among others, by M. Kurihara, A. Zinoviev and S. Vostokov. K. Kato also has explicit formulas for the higher-dimensional Kummer pairing associated to certain (one-dimensional) $p$-divisible groups. In this paper we construct an explicit reciprocity law describing the Kummer pairing associated to any (one-dimensional) formal group. The formulas are a generalization to higher-dimensional local fields of Kolyvagin's reciprocity laws. The formulas obtained describe 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. In the second part of this paper, we will apply the results obtained here to give explicit formulas for the generalized Hilbert symbol and the Kummer pairing associated to a Lubin-Tate formal group. The results obtained in the second paper constitute a generalization to higher local fields, of the formulas of Artin-Hasse, K. Iwasawa and A. Wiles., The stronger reciprocity laws in this new version cover arbitrary higher local fields, as opposed to only standard higher local fields

To give a unified treatment on the association of Lie algebras and vertex algebras, we study ( G , χ ϕ ) -equivariant ϕ-coordinated quasi modules for vertex algebras, where G is a group with χ ϕ a linear character of G and ϕ is an associate of the one-dimensional additive formal group. The theory of ( G , χ ϕ ) -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 , χ ϕ ) -equivariant ϕ-coordinated quasi modules. Furthermore, for any conformal algebra C , we construct a class of Lie algebras C ˆ ϕ [ G ] and prove that restricted C ˆ ϕ [ G ] -modules are exactly ( G , χ ϕ ) -equivariant ϕ-coordinated quasi modules for the universal enveloping vertex algebra of C . As an application, we determine the ( G , χ ϕ ) -equivariant ϕ-coordinated quasi modules for affine and Virasoro vertex algebras.

We explicitly construct and investigate a number of examples of $\mathbb{Z}/p^r$-equivariant formal group laws and complex-oriented spectra, including those coming from elliptic curves and $p$-divisible groups, as well as some other related examples., A first version of this article was written in 2018. The current version, which includes some new material, was accepted for publication in the JHRS

In this paper we study the $p$-adic dynamics of prime-to-$p$ Hecke operators on the set of points of modular curves in both cases of good ordinary and supersingular reduction. We pay special attention to the dynamics on the set of CM points. In the case of ordinary reduction we employ the Serre-Tate coordinates, while in the case of supersingular reduction, we use a parameter on the deformation space of the unique formal group of height $2$ over $\overline{\mathbb{F}}_p$, and take advantage of the Gross-Hopkins period map.

We construct a tower of fields from the rings $R_n$ which parametrize pairs $(X,\lambda)$, where $X$ is a deformation of a fixed one-dimensional formal group $\mathbb{X}$ of finite height $h$, together with a Drinfeld level-$n$ structure $\lambda$. We choose principal prime ideals $\mathfrak{p}_n \mid (p)$ in each ring $R_n$ in a compatible way and consider the field $K'_n$ obtained by localizing $R_n$ at $\mathfrak{p}_n$, completing, and passing to the fraction field. By taking the compositum $K_n = K'_n K_0$ of each field 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 Anthony Scholl. When $h=2$ we also investigate the question of whether this is a Kummer tower.

We prove $p$-adic versions of a classical result in arithmetic geometry stating that an irreducible subvariety of an abelian variety with dense torsion has to be the translate of a subgroup by a torsion point. We do so in the context of certain rigid analytic spaces and formal groups over a $p$-adic field $K$ or its ring of integers $R$, respectively. In particular, we show that the rigidity results for algebraic functions underlying the so-called Manin-Mumford Conjecture generalize to suitable $p$-adic analytic functions. In the formal setting, this approach leads us to uncover purely $p$-adic Manin-Mumford type results for formal groups not coming from abelian schemes. Moreover, we observe that a version of the Tate-Voloch Conjecture holds in the $p$-adic setting: torsion points either lie squarely on a subscheme or are uniformly bounded away from it in the $p$-adic distance.

In a recent article, the authors established an explicit description of kernels in the category of the formal group laws over the ring of Witt vectors over a finite field in terms of Fontaine’s triples. The present research is devoted to an adjacent problem of explicit description of cokernels. The technique developed is applied to a natural monomorphism from $$F_m$$ to the Weil restriction of $$F_m$$ with respect to certain ring extensions. Besides, we investigate some properties of the category of formal group laws over the ring of Witt vectors such as left and right integrability and left and right semi-abelianity.

The variation in the irregularity degree of a finite unramified local field extensions of a local field is investigated with respect to a polynomial formal group and in the multiplicative case. The necessary and sufficient conditions for the existence of the psth primitive roots of the psth power of 1 and (endomorphism $${{[{{p}^{s}}]}_{{{{F}_{m}}}}}$$ ) in the Lth unramified extension of the local field K (for all positive integers s) are found. The conditions depend only on the ramification index of the maximal Abelian subextension of the field K Ka/ $${{\mathbb{Q}}_{p}}$$ .

Generalized Honda formal groups are a class of formal groups, which includes all formal groups over the ring of integers of local fields weakly ramified over $${{\mathbb{Q}}_{p}}$$ . This class is the next in the chain multiplicative formal group–Lubin-Tate formal groups–Honda formal groups. The Lubin-Tate formal groups are defined by distinguished endomorphisms [π]F. Honda formal groups have distinguished homomorphisms that factor through [π]F. In this article, we prove that for generalized Honda formal groups, the composition of a sequence of distinguished homomorphisms factors through [π]F . As an application of this fact, a number of properties of πn-torsion points of the generalized Honda formal group are proved.

Rapoport–Zink spaces are deformation spaces for $p$-divisible groups with additional structure. At infinite level, they become preperfectoid spaces. Let ${{\mathscr M}}_{\infty }$ be an infinite-level Rapoport–Zink space of EL type, and let ${{\mathscr M}}_{\infty }^{\circ }$ be one connected component of its geometric fiber. We show that ${{\mathscr M}}_{\infty }^{\circ }$ contains a dense open subset which is cohomologically smooth in the sense of Scholze. This is the locus of $p$-divisible groups which do not have any extra endomorphisms. As a corollary, we find that the cohomologically smooth locus in the infinite-level modular curve $X(p^{\infty })^{\circ }$ is exactly the locus of elliptic curves $E$ with supersingular reduction, such that the formal group of $E$ has no extra endomorphisms.

The Lubin-Tate moduli space X0rig is a p-adic analytic open unit polydisc which parametrizes deformations of a formal group H0 of finite height defined over an algebraically closed field of characteristic p. It is known that the natural action of the automorphism group Aut(H0) on X0rig gives rise to locally analytic representations on the topological duals of the spaces H0(X0rig , (ℳ0 )rig) of global sections of certain equivariant vector bundles (ℳ0 )rig over X0rig . In this article, we show that this result holds in greater generality. On the one hand, we work in the setting of deformations of formal modules over the valuation ring of a finite extension of ℚp. On the other hand, we also treat the case of representations arising from the vector bundles (ℳ )rig over the deformation spaces Xrig with Drinfeld level-m-structures. Finally, we determine the space of locally finite vectors in H0(Xrig , (ℳ )rig). Essentially, all locally finite vectors arise from the global sections of invertible sheaves over the projective space via pullback along the Gross-Hopkins period map.

A global analog of Fontaine’s category of triples is introduced. While the latter is antiequivalent to the category of formal group laws over the ring of Witt vectors over a finite field of charact...

The main fields of application of formal groups are algebraic geometry and class field theory. The later uses both the classical Hilbert symbol (the norm-residue symbol) and its generalization. One of the most important problems is finding explicit formulas for various modifications of this symbol related to formal groups. There are two approaches to constructing formal groups (i.e., power series satisfying certain conditions). The functional lemma proved by Hazewinkel allows one to make formal groups with coefficients from a ring of characteristic zero by means of functional equations using a certain ideal of this ring, an overfield of the ring, and a ring homomorphism with specified properties (e.g., identical; for a local field, the Frobenius homomorphism can be chosen). There is a convenient criterion for the isomorphism of formal groups, constructed by Hazewinkel’s formula, as well as a formula for logarithms (in particular, the Artin–Hasse logarithm). At the same time, Lubin and Tate construct formal groups over local fields, using isogeny, and Honda, when constructing formal groups over the ring of integers of a discrete valued field of characteristic zero, introduces a certain noncommutative ring induced by the original ring and a fixed homomorphism. The paper relates the classical classification of formal groups (standard, generalized, relative Lubin–Tate formal groups, and formal Honda groups) to their classification using the Hazewinkel functional lemma. For each type, the corresponding functional equations are composed and logarithms, as well as series used to construct an explicit formula for the Hilbert symbol, are studied.

In this paper, various extensions of local fields are considered. For arbitrary finite extension K of the field of p-adic numbers, the maximum Abelian extension KAb/K and the corresponding Galois group can be described using the well-known Lubin–Tate theory. It is represented as a direct product of groups obtained using the maximum unramified extension of K and a fully ramified extension obtained using the roots of some endomorphisms of Lubin–Tate formal groups. We consider the so-called “generalized Lubin–Tate formal groups” and extensions obtained by adding the roots of their endomorphisms to the field under consideration. Using the fact that a correctly chosen generalized formal group coincides with the classical one over unramified finite extension Tm of degree m of field K, it was possible to obtain the Galois group of the extension (Tm)Ab/K. The main result of the work, is an explicit description of the Galois group of the extension (Kur)Ab/K, where Kur is the maximum unramified extension of K. Similar methods are also used to study ramified extensions of the field K.

We import into homotopy theory the algebrogeometric construction of the cotangent space of a geometric point on a scheme. Specializing to the category of spectra local to a Morava [math] –theory of height [math] , we show that this can be used to produce a choice-free model of the determinantal sphere as well as an efficient Picard-graded cellular decomposition of [math] . Coupling these ideas to work of Westerland, we give a “Snaith’s theorem” for the Iwasawa extension of the [math] –local sphere.

We present five open problems in the theory of vertex rings. They cover a variety of different areas of research where vertex rings have been, or are threatening to be, relevant. They have also been chosen because I personally find them interesting, and because I think each of them has a chance (the title of the paper notwithstanding!) of being solved. In each case we give some explanatory background and motivation, sometimes including proofs of special cases. Beyond vertex rings per se, the topics covered include connections to real Lie theory, formal group laws, modular linear differential equations, Pierce bundles, and genus 2 Siegel modular forms and the Moonshine Module.

In this paper we investigate the irregular degree of finite not ramified local field extantions with respect to a polynomial formal group and in the multiplicative case. There was found necessary and sufficient conditions for the existence of primitive roots of ps power from 1 and (endomorphism [ps]Fm) in L-th unramified extension of the local field K (for all positive integer s). These conditions depend only on the ramification index of the maximal abelian subextension of the field K Ka/Qp.

Generalized Honda formal groups are a new class of formal groups that in particular describes the formal groups over the ring of integers of local fields weakly ramified over Qp. It is the next class in the chain the multiplicative formal group — Lubin — Tate formal groups — Honda formal groups. Lubin — Tate formal groups are defined by distinguished endomorphisms [π]F , Honda formal groups possess distinguished omomorphisms that factor through [π]F and in the present paper we prove that for generalized Honda formal groups it is compositions of distinguished homomorphisms that factor through [π]F . As an application of this fact, some properties of πn-torsion points of generalized Honda formal groups are studied.

The main purpose of this article is to define a quadratic analog of the Chern character, the so-called Borel character, which identifies rational higher Grothendieck-Witt groups with a sum of rational MW-motivic cohomologies and rational motivic cohomologies. We also discuss the notion of ternary laws due to Walter, a quadratic analog of formal group laws, and compute what we call the additive ternary laws, associated with MW-motivic cohomology. Finally, we provide an application of the Borel character by showing that the Milnor-Witt K-theory of a field F embeds into suitable higher Grothendieck-Witt groups of F modulo explicit torsion., Comment: Major changes. Comments are still welcome!

Статья посвящена памяти Георгия Вороного. Описываются новые избранные результаты о рядах Эйзенштейна, о (мотивных), (p-адических), (кратных) значениях (круговых) дзета и L-функций, и их приложения, полученные ниже перечисляемыми авторами, а также элементарное введение в эти результаты. Дан краткий обзор новых результатов о (мотивных), (p-адических), (кратных) значениях (круговых) дзета функциях, L-функциях и рядах Эйзенштейна. Статья ориентирована на избранные задачи и не является исчерпывающей. Начало статьи содержит краткое изложение результатов о числах Бернулли, связанных с исследованиями Георгия Вороного. Результаты о кратных значениях дзета функций были представлены Д. Загиром, П. Делинем и А. Гончаровым, А. Гончаровым, Ф. Брауном, К. Глэносом (Glanois) и другими. С. Унвер ("Unver) исследовал кратные p-адические дзета-значения глубины два. Таннакиева интерпретация кратных p-адических дзета-значений дана Х. Фурушо. Краткая история и связи между группами Галуа, фундаментальными группами, мотивами и арифметическими функциями представлены в докладе Ю. Ихара. Результаты о кратных дзета-значениях, группах Галуа и геометрии модулярных многообразий представлены Гончаровым. Интересная унипотентная мотивная фундаментальная группа определена и исследована Делинем и Гончаровым. В данной работе мы кратко упоминаем в рамках (p-адических) L-функций и (p-адических) (кратных) дзета-значений применения подходов Куботы-Леопольдта и Ивасавы, которые основанны на p-адических L-функциях Куботы-Леопольда, и арифметических p-адических L-функциях Ивасавы. Прореферирован ряд недавних работ (и соответствующих результатов): кратные дзета-значения в корнях из единицы, построение семейств мотивных итерированных интегралов с предписанными свойствами по Глэносу (Glanois); явные выражения для круговых p-адических кратных дзета-значений глубины два по Унверу (Unver); связи арифметических степеней циклов Кудлы-Рапопорта на интегральной модели многообразия Шимуры, соответствующей унитарной группе сигнатуры (1,1), с коэффициентами Фурье центральных производных рядов Эйзенштейна рода 2 по Санкарану (Sankaran). Более полно с содержанием статьи можно ознакомиться по приводимому ниже оглавлению: Введение. 1. Сравнения типа Вороного для чисел Бернулли. 2. Римановы дзета-значения. 3. О группах классов колец с теорией дивизоров. Мнимые квадратичные и круговые поля. 4. Ряды Эйзенштейна. 5. Группы классов, поля классов и дзета-функции. 6. Кратные дзета-значения. 7. Элементы неархимедовых локальных полей и неархимедова анализа. 8. Итерированные интегралы и (кратные) дзета-значения. 9. Формальные и p-делимые группы. 10. Мотивы и (p-адические) (кратные) дзета-значения. 11. О рядах Эйзенштейна, ассоциированных с многообразиями Шимуры. Разделы 1-9 и подраздел 11.1 (О некоторых многообразиях Шимуры и модулярных формах Зигеля) можно рассматривать как элементарное введение в результаты раздела 10 и подраздела 11.2 (О несобственном пересечении дивизоров Кудлы-Рапопорта и рядах Эйзенштейна).Я глубоко признателен Н. М. Добровольскому за помощь и поддержку в процессе подготовки статьи к печати.

New results are presented and a brief review is given for new methods of the theory of dynamic systems on manifolds over local fields and formal groups over local rings. For the analysis of n-dimensional manifolds and dynamic systems on such manifolds, formal structures are used, in particular, n-dimensional formal groups. Infinitesimal deformations are presented in terms of formal groups. The well-known one-dimensional case is extended and n-dimensional (n ≥ 1) analytic mappings of an open p-adic polydisc (n-disk) $$ {D}_p^n $$ are considered. The n-dimensional analogs of modules arising in formal and non-Archimedean dynamic systems are introduced and investigated and their formal-algebraic structure is presented. Rigid structures, objects, and methods are outlined. From the point of view of systems analysis, new, namely formal and non-Archimedean, faces and structures of systems, mappings and iterations of mappings between these faces and structures are introduced and investigated.

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.

A complete description of Buchstaber formal groups $$F(u,v)=\frac{u^{2}A(v)-v^{2}A(u)}{uB(v)-vB(u)},$$ in which the series A(x) and B(x) satisfy the relation A(x)l = B(x)m, is given. A new family of Buchstaber formal groups depending on two algebraically independent parameters is obtained.

We present formulas for the components of the Buchstaber formal group law and its exponent over ℚ[p1, p2, p3, p4]. This leads to an addition theorem for the general elliptic integral $$\int_0^x {dt{\rm{/}}R\left( t \right)} $$ with $$R(t)=\sqrt{1+p_{1} t+p_{2} t^{2}+p_{3} t^{3}+p_{4} t^{4}}$$ . The study is motivated by Euler’s addition theorem for elliptic integrals of the first kind.

We present a formal power series ∑ A i ⁢ j ⁢ x i ⁢ y j {\sum A_{ij}x^{i}y^{j}} over the Lazard ring Λ and the formal group laws F n {F_{n}} , n ≥ 2 {n\geq 2} , over the quotient rings of Λ. For each F n {F_{n}} , we construct a complex cobordism theory with singularities with the coefficient ring ℚ ⁢ [ p 1 , … , p 2 ⁢ n ] {\mathbb{Q}[p_{1},\dots,p_{2n}]} , with parameters p i {p_{i}} , | p i | = 2 ⁢ i {|p_{i}|=2i} .

The structure of the first Galois cohomology groups for the group of points of a formal module in extensions of local fields is studied. A complete description for unramified extensions and classical formal group laws is obtained.

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.

We describe additive (unstable) operations from a theory A^* obtained from Algebraic Cobordism of M.Levine-F.Morel by change of coefficients to any oriented cohomology theory B^*. We prove that there is 1-to-1 correspondence between the set of operations, and the set of transformations: A^n((P^{\infty})^{\times r}) ---> B^m((P^{\infty})^{\times r}) satisfying certain simple properties. This provides an effective tool of constructing such operations. As an application, we prove that (unstable) additive operations in Algebraic Cobordism are in 1-to-1 correspondence with the L\otimes_Z Q-linear combinations of Landweber-Novikov operations which take integral values on the products of projective spaces. On our way we obtain that stable operations there are exactly L-linear combinations of Landweber-Novikov operations. We also show that multiplicative operations A^* ---> B^* are in 1-to-1 correspondence with the morphisms of the respective formal group laws. We construct Integral (!) Adams Operations in Algebraic Cobordism, and all theories obtained from it by change of coefficients, giving classical Adams operations in the case of K_0. Finally, we construct Symmetric Operations for all primes p (these operations in Algebraic Cobordism, previously known only for p=2, are more subtle than the Landweber-Novikov operations, and have applications to rationality questions), as well as the T.tom Dieck - style Steenrod operations in Algebraic Cobordism. As a bi-product of the proof of our main theorem we get the Riemann-Roch Theorem for additive (unstable) operations., to appear in Annales Scientifiques de l'Ecole Normale Superieure