p-адические L-функции и p-адические кратные дзета значения

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.


Introduction
The article is dedicated to the memory of George Voronoi. It is concerned with ( -adic)functions (in partially ( -adic) zeta functions) and cyclotomic ( -adic) (multiple) zeta values. The beginning of the article contains a short summary of the results on the Bernoulli numbers associated with the studies of George Voronoi. Results on multiple zeta values have presented by D. Zagier [1], by P. Deligne and A.Goncharov [5], by A. Goncharov [6], by F. Brown [7], by C. Glanois [8] and others. Tannakian interpretation of -adic multiple zeta values is given by H. Furusho [10]. Short history and connections among Galois groups, fundamental groups, motives and arithmetic functions are presented in the talk by Y. Ihara [12]. Results on multiple zeta values, Galois groups and geometry of modular varieties has presented by Goncharov [6]. Interesting unipotent motivic fundamental group is defined and investigated by Deligne and Goncharov [5]. S.Ünver [9,11] have investigated -adic multiple zeta values in the depth two. The framework of ( -adic) -functions and ( -adic) (multiple) zeta values is based on Kubota-Leopoldt -adic -functions [13] and arithmeticadic -functions by Iwasawa [14]. Motives and ( -adic) (multiple) zeta values, improper intersections of Kudla-Rapoport divisors and Eisenstein series by Sankaran [37] are reviewed. More fully the content of the article can be found at the following table of contents: Introduction. 1. Voronoi-type congruences for Bernoulli numbers. 2. Riemann zeta values. 3. On class groups of rings with divisor theory. Imaginary quadratic and cyclotomic fields. 4. Eisenstein Series. 5. Class group, class fields and zeta functions. 6. Multiple zeta values. 7. Elements of non-Archimedean local fields and −adic analysis. 8. Iterated integrals and (multiple) zeta values. 9. Formal groups and -divisible groups. 10. Motives and ( -adic) (multiple) zeta values. 11. On the Eisenstein series associated with Shimura varieties. Sections 1-9 and subsection 11.1 (On some Shimura varieties and Siegel modular forms) can be considered as an elementary introduction to the results of section 10 and subsection 11.2 (On improper intersections of Kudla-Rapoport divisors and Eisenstein series). Numerical examples are included.
The subject matter of this review has deep historical roots, with contributions of many mathematiciens. I apologize for any oversights and any misrepresentations, which are not intentional but rather due to my ignorance.  [2,3]. Author of the text attended the lectures and seminars of Yu. Manin. Following of the kind conversation with Yu. Manin the author has implemented the computer program and has computed Manin's modular symbols [39] for elliptic curve Γ 0 (11)

Voronoi's congruences
Let be an natural number (the modulus), coprime with and let 2 = 2 2 be the Bernoulli number with coprime 2 and 2 . Then

Kummer congruences
If is prime and − 1 not divide even positive then the number is -integer and there is the congruence

Riemann zeta values
Here we follow to [15,16,17,18]. Let = + be a complex number and let ( ) be the Riemann zeta function which is presented for > 1 by the series By Euler for 1 where 2 are Bernoulli numbers; recall also that for odd = 1, 3, 5, . . ..
3. On class groups of rings with divisor theory. Imaginary quadratic and cyclotomic fields The study of class groups of rings and corresponding schemes is an actual scientific problem (see [18,20] and references therein). For regular local rings, according to the Auslander-Buchsbaum theorem, the (divisors) class group is trivial. But in most interesting cases the group is nontrivial. The Heegner approach, together with the results of Weber, Birch, Baker and Stark, makes it possible to calculate and even parametrize rings with a given (small) class number in some cases. Let be a commutative ring with identity for which there exists the theory of divisors [18]. The order of the class group is calculated on the basis of the use of -functions. We investigate one of the aspects of this problem, consisting in finding the moduli spaces of elliptic curves defined over the rings with the given class number.
Problem. To investigate the case of elliptic curves over rings of integers of quadratic fields (rings of integers of quadratic algebraic extensions of the field of rational numbers Q) with a small class number, see [18].
In some cases, for instance under computer algebra computations, we have to enumerate investigated objects. Some simple parametric spaces and moduli spaces in the case of imaginary quadratic fields are presented below [40]. We present an elementary introduction to this problem and give the moduli spaces as trivial bundles over affine part of the groups of rational points of some elliptic curves over the ring of integers Z. Below we present parameter spaces and moduli for class number one and two. Let  Let 2 be the affine part of the group of rational points over Z of the elliptic curve 3 + 3 = − 2 , let 3 be the affine part of the group for the elliptic curve 3 − 3 = 2 2 , and Proposition 4. Let be the imaginary quadratic field with class number two. Then the moduli spaces of elliptic curves of the form (*), without an exceptional case, are the trivial bundles

Eisenstein Series
Here we follow to [15,16,17,18]. Let belong to the modular figure of the modular group Γ = Γ(1). Definition 1. In these notations with > 1 the Eisenstein series is defined as Proposition 6. Eisenstein series have the representation If we will use functions of the sums of divisors 2 −1 we obtain Put 2 = 60 2 , 3 = 140 3 .
Let us transform in such a way that corresponding Fourier coefficients under , 1 will rational numbers. Dividing on 2 (2 ) and denoting the obtained result as we have by the Corollary 1

Class group, class fields and zeta functions
Here we follow to [16,18]. Let be an imaginary quadratic field and let be its class group. Definition 3. Let (a) be the norm of the ideal a. The Dedekind -function for is defined for all > 1 by the series where the sum is taken over all nonzero ideals a ∈ .
Let be a subring ( ̸ = Z) of the ring of integers of the imaginary quadratic field . Let 1 , . . . ℎ be pairwise nonequivalent modules of with the same ring of multipliers . Proposition
Follow to [16] it is possible to define ray class field. As in an imaginary quadratic field there is no real infinite primes so modulus of the field is an ideal of the ring of integers of the field.
Let m be a modulus of the an imaginary quadratic field , let m be the ray class group, let be the Weber function . Let R ∈ and let R * ∈ m be the ideal class whose image in is equal to (m)R −1 .
Let be an ideal class. Below we present values of zeta and -functions connecting with imaginary quadratic fields. Let be a squarefree integer number, = Q( √ ) a quadratic field, be the character of the quadratic field . Let ( , ) be the −series with a nonunit character modulo | |. Here is the discriminant of the field .
Let be the number of roots of unity of the imaginary quadratic field [ : Q].
for all other imaginary quadratic fields.
Let ℎ be the class number of the field . Proposition 14. ( Corollary 2. For imaginary quadratic fields with class number one (ℎ = 1) we have The multiple zeta value of the weight and the depth is called the expression of the form Let be the group of roots of unity. Definition 7. Let 1 , . . . be natural numbers with 2. The multiple zeta value relative to of the weight and the depth is called the expression of the form

Elements of non-Archimedean local fields and −adic analysis
Here we present elements of −adic local fields, their algebraic extensions and −adic interval analysis. We follow to [18,21].

Elements of non-Archimedean local fields
A non-Archimedean local field is a complete discrete valuation field with finite residue field. Further, for brevity, we call these fields local. In other words, a field is called local if it is complete in a topology determined by the valuation of the field and if its residue field is finite. We assume further that the valuation is normalized, i.e. the homomorphism of the multiplicative group of the field to the additive group of rational integers : * → Z is surjective.
The structure of such fields is known: if the field has the characteristic zero, then it is a finite extension of the −adic field Q , which is the completion of the field of rational numbers with respect to the −adic valuation.
If Denote by the maximal unramified extension of the field (in a fixed algebraic closure of the field ) with a residue field , which is the algebraic closure of a field .
In a non-Archimedean local field each of its elements has a representation = , where is a unit of the ring of integers of the field and its uniformizing element, that is ( ) = 1 , is an integer rational number. A unit is called principal if ≡ 1 (mod ). Let = 1 ∪ 2 ∪ · · · ∪ be a partition of ⊂ . Recall that the additive mapping of a set of compact-open subsets of with value in Q is called the −adic distribution on : ( ) = ( 1 ) + ( 2 ) + · · · + ( ).

Bernoulli distributions.
Let ( ) be the −Bernoulli polynomial. These polynomials are defined by the decomposition We have:

Iterated integrals and (multiple) zeta values
Here we follow to [22,23]. Let C be the complex plane and ( ) be the holomorphic function on C . Let ( ) be the differential of the first kind on C. Let be a Riemann surfaces and be the differential of the first kind on . Parshin has considered iterated integrals of this type on Riemann surfaces [22]. Chen [23] for smooth paths on a manifold and respective path spaces have investegated iterated (path) integrals. For differential forms 1 , . . . , on he has constructed the iterated integrals by repeating times the integration of the path space differential forms (and their linear combinations). Chen [23] has denoted the iterated integrals as ∫︀ 1 2 · · · and set ∫︀ 1 2 · · · = 1 when = 0 and ∫︀ 1 2 · · · = 0 when < 0.

Remark 11.
(2) = More generally iterated integrals are path space differential forms which permit further integration.

Formal groups and -divisible groups
Recall some definitions. Let be a complete discrete variation field with the ring of integers and the maximal ideal . A complete discrete variation field with finite residue field is called a local field [24]. A complete discrete variation field with algebraically closed residue field is called a quasi-local field [26]. Below we will suppose that in the case the characteristic of satisfies > 0. Let be a local or quasi-local field. If is a local field [24] and has the characteristic 0 then it is a finite extension of the field of -adic numbers Q . Let be the normalized exponential valuation of . If [ : Q ] = then = · , where = ( ) and = [ : F ], where is the residue field of (always assumed perfect ). If has the characteristic > 0 then it isomorphic to the field (( )) of formal power series, where is uniformizing parameter. Let be a finite extension of a local field , , their residue fields, = ℎ and / ramification index of over . An extension / is said to be unramified if / = 1 and extension / is separable. An extension / is said to be tamely ramified if not devides / and the residue extension / is separable. An extension / is said to be totally ramified if / = [ : ] = ( ℎ ) , 1. Let / be the finite Galois extension of quasi-local field with Galois group , ( , ) one dimensional formal group low over the ring of integers of the field , ( ) be the -module, that is defined by the group low ( , ) on the maxilal ideal of the ring , ( ∈ Z, 1) be the subgroup of -th degrees of elements from , := ( ). Below we will suppose that ℎ > 3.

Norm Maps
Here we use results on formal groups from [27,25]. Let = ( ) be the -module that is defined by the -dimentional group low ( , ) on the product ( ) := × · · · × , ( times) of maximal ideals of the ring of any finite Galois extension of the field . Let := ℎ , := ( ), ( = +∞, if characteristic of the field is equal and is positive integer in the opposide case), / be the Galois extension of the prime degree , ( , ) be the one dimensional group low over . Let := ℎ > 0.
are coefficients of the -iteration of the group low.
Let be a commutative ring. Let , , be finite group schemes over . The sequence Let be a prime number and ℎ be an integer, ℎ 0. Recall the definition of the -divisible group by J. Tate. Definition 11. A -divisible group over of height ℎ is an inductive system is a finite group scheme over of order ℎ , (ii) for each 0, is exact. Let = Q( ) be the cyclotomic field, ∈ be a primitive th root of unity and be the ring of integers of . The corresponding multiple zeta values at arguments ∈ N, ∈ can be expressed in terms of the coefficients of a version of Drinfeld's assosiators by Drinfeld [28], which in turn, can be expressed in terms of periods of the corresponding motivic multiple zeta values (MMZV).
The algebra ℋ carries an action of the motivic Galois group of the category of mixed Tate motives over [1/ ]. The author studies the Galois action on the motivic unipotent fundumental groupoid of P 1 ∖{0, , ∞} (or of G ∖ ) for next values of : His results include: bases of multiple zeta values via multiple zeta values at roots of unity for the above ; more generally, constructing of families of motivic iterated integrals with prescribed properties; the new proof, via the coproduct by Goncharov [29] and its extension by Brown [7], of the results by Deligne [30] that the Tannakian category of mixed Tate motives over [1/ ] 'for = {2, 3, 4, 8} is spanned by the motivic fundumental groupoid of P 1 ∖{0, , ∞} with an explicit basis'.
In article [11]Ünver continues his investigation of -adic multiple zeta values [9], presenting a computation of values of the p-adic multiple polylogarithms at roots of unity. The main result of the paper [11] (Theorem 6.4.3 with Propositions 6.4.1 and 6.3.1) is to give explicit expression for the cyclotomic -adic multi-zeta values ( 1 , 2 ; 1 , 2 ) of depth two. The result is far too technical to state here.
The proof of the theorem is rather technical; it is based on rigid analytic function arguments and a long distance analysis of group-like elements of related algebras.
For number fields the category of realizations has defined and investigated by Deligne [4]. Results on multiple zeta values, Galois groups and geometry of modular varieties has presented by Goncharov [6]. Interesting unipotent motivic fundamental group is defined and investigated by Deligne and Goncharov [5]. Tannakian interpretation of -adic multiple zeta values is given by Furusho [10].
Results obtained in the paper [11] may be applied to the problems of the -adic theory of higher cyclotomy.

On the Eisenstein series associated with Shimura varieties
Interesting classes of Shimura varieties form varieties which have an interpretation as moduli spaces of abelian varieties. Moduli spaces of corresponding −divisible groups over perfect fields of characteristic are used for investigation of the local structure of these Shimura varieties. Denote by Π the period matrix of the abelian variety . This is × 2 complex matrix Π = ( 1 , 2 ) with nondegenerate × matrices 1 and 2 . Definition 14. The period matrix Π is called normalized if it has the form ( , ) where is the unit × matrix and ∈ H , where.
is the Siegel upper half-plane. Here is the matrix transposed to . belong to Γ (its determinant is equal 1) and so ( ) has a representation by the Fourier series.

On improper intersections of Kudla-Rapoport divisors and Eisenstein series
Let be an imaginary quadratic field, its ring of integers and , be the ring of integers of the completion of at . Sankaran [37] proves that the arithmetic degrees of Kudla-Rapoport cycles on an integral model of a Shimura variety attached to a unitary group of signature (1, 1) are Fourier coefficients of the central derivative of an Eisenstein series of genus 2. The main results of the paper are the following Theorem 4.13 on the value of the Eisenstein series and the Corollary 4.15 on the relation between the arithmetic degree of special cycle and the Eisenstein series. These results confirm conjectures by Kudla [31] and by Kudla, Rapoport [32] on relations between intersection numbers of special cycles and the Fourier coefficients of automorphic forms in the degenerate setting and for dimension 2. As have pointed out by Kudla [33] and others 'these relations may be viewed as an arithmetic version of the classical Siegel-Weil formula, which identifies the Fourier coefficients of values of Siegel-Eisenstein series with representation numbers of quadratic forms'. In the paper by Kudla, Rapoport [34] 'the Shimura variety is replaced by a formal moduli space of -divisible groups, the special arithmetic cycles are replaced by formal subvarieties, and the special values of the derivative of the Eisenstein series are replaced by the derivatives of representation densities of hermitian forms.' Sankaran defines the local Kudla-Rapoport cycles ( ) and gives some applications of results obtained in his earlier paper [38] where he proved the Theorem 3.14 on cycles ( ). He allows 'the polarizations to be non-principal in a controlled way'. An unpolarized case of -divisible groups with the given -kernel type and with applications to their Newton polygons has considered in the paper by Harashita [35]. Sankaran's paper [37] consists of four sections. The first section presents the purpose of the paper and short description of ideas and results of next sections. Second section concerns with local Kudla-Rapoport cycles on the Drinfeld upper half-plane. The main result of this section is the Theorem 2.14 on values of local intersection numbers of these cycles. The third section is devoted to the prove of the closed-form formula for representation densities ( , ). Author specializes the explicit formula on Hermitian representation densities by Hironaka [36] to the case at hand: ( , , ) ∈ [ ], ∈ 2 ( , ), ( ) is even, = ( , 1), ( , ) = ( , , (− ) − ). In the last section global aspects are discussed and main result is presented. Let (1,1) denote the Deligne-Mumford (DM) stack over of almostprincipally polarized abelian surfaces and ℰ the DM stack over of principally polarized elliptic curves with multiplication by . In conditions of the subsection 4.1 of the paper [37] author sets ℳ = ℰ × (1,1) and define for ∈ 2 ( ) cycles ( ). Then in subsection 4.3 the author prove Theorem 4.13 and Corollary 4.15.

Conclusions
Classical and novel results on ( -adic) -functions, ( -adic) (multiple) zeta values and Eisenstein series are presented.