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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 21. Выпуск 3.
УДК 511+513.6+519.45 DOI 10.22405/2226-8383-2020-21-3-68-83
О гипотезе Ленглендса, глобальных полях и (Д)-штуках
H. М. Глазунов
Николай Михайлович Глазунов — доктор физико-математических наук, профессор, Национальный авиационный университет (г. Киев, Украина). e-mail: glanmMyahoo.com,
Аннотация
В обзоре, который посвященн 80-десятилетию A.B. Яковлева, 75-петию C.B. Востокова и 75-летию В.В Лурье, представлены избранные результаты реализации программы Ленглендса над глобальными полями. Работы юбиляров связаны с алгебраической теорией чисел в её как локальных, так и глобальных аспектах, и с построением соответствующих теорий полей классов. Гипотезы Ленглендса, как отметил И.Р. Шафаревич, имеют целью "обобщение теории полей классов, аналогичное обобщению теории абелевых функций". Обзор является введением в программу Ланглендса, глобальные поля, Д-штуки и конечные штуки над полями функцый алгебраических кривых над конечными полями, и не является исчерпывающим.В зависимости от выбора основного поля, результаты реализации программы Ленглендса были получены и обсуждались Ленглендсом, Жаке, Шафа-ревичем, Паршиным, Дринфельдом, Лаффорге и другими. Напомним, что линейные алгебраические группы нашли важные приложения в программе Ланглендса. Именно, для связной редуктивной группы G над глобальным полем К соответствие Ленглендса соотносит автоморфные формы на G и глобальные параметры Ленглендса, а именно, классы сопряженности гомоморфизмов из абсолютной группы Галуа поля К в группу Ленглендса LG. Для полей алгебраических чисел применения и развитие программы Ленглендса позволило усилить теорему Вайлса о гипотезе Шимуры-Таниямы-Вейля и доказать гипотезу Сато-Тейта. Дринфельд и Лаффорге исследовали случай общей линейной группы над глобальным функциональными полями ненулевой характеристики (Дринфельд для G = GL2 и Лаффорге для GLr, г произвольное положительное целое) и доказали в этом случае соответствие Ленглендса. В процессе этих исследований Дринфельдом была введена концерсия ^-пучков, или штук, которая использовалась обоими авторами в процессе установления соответствия Ленглендса. Наряду с использованием штук, были предложены и использованы другие конструкции. Андерсен предложил концепспю t-мотива. Хартль, его коллеги и ученики предложили и исследовали (связанные со штуками, t-мотивами и ^-пучками) копцепсии конечных, локальных и глобальных С-штук. В предлагаемой обзорной статье мы начинаем с краткого представления результатов программы Ленглендса над полями алгебраических чисел и их локализаций. Далее кратко представлены подходы Хартля, его коллег и учеников. Эти подходы и их обсуждение связаны как с программой Ленглендса, так и с внутренним развитием теории G-штук.
Автор признателен анонимному рецензенту за замечания и советы, доктору Зиян Дингу (Zhiyuan Ding) за замечание, U.M. Добровольскому за помощь и поддержку в процессе подготовки статьи к печати.
Ключевые слова: соответствие Ленглендса, глобальное поле, модуль Дринфельда, штука, конечная штука, локальный модуль Андерсона, кокасательный комплекс, формальная группа.
Библиография: 48 названий.
Для цитирования:
Н. М. Глазунов. О гипотезе Ленглендса, глобальных полях и (Д)-штуках // Чебышевский сборник, 2020, т. 21, вып. 3, с. 68-83.
CHEBYSHEVSKII SBORNIK Vol. 21. No. 3.
UDC 511+513.6+519.45 DOI 10.22405/2226-8383-2020-21-3-68-83
On Langlands program, global fields and shtukas
N. M. Glazunov
Nikolay Mihaylovich Glazunov — doctor of physical and mathematical Sciences, Professor, National Aviation University (Kiev, Ukraine). e-mail: glanmMyahoo.com,
Abstract
The purpose of this paper is to survey some of the important results on Langlands program, global fields, ^-shtukas and finite shtukas which have influenced the development of algebra and number theory. It is intended to be selective rather than exhaustive, as befits the occasion of the 80-th birthday of Yakovlev, 75-th birthday of Vostokov and 75-th birthday of Lurie. Under assumptions on ground fields results on Langlands program have been proved and discussed by Langlands, Jacquet, Shafarevich, Parshin, Drinfeld, Laflbrgue and others.
This communication is an introduction to the Langlands Program, global fields and to ^-shtukas and finite shtukas (over algebraic curves) over function fields. At first recall that linear algebraic groups found important applications in the Langlands program. Namely, for a connected reductive group G over a global field K, the Langlands correspondence relates automorphic forms on G and global Langlands parameters, i.e. conjugacy classes of homomorphisms from the Galois group Qal(K/K) to the dud Langlands group G(Qp). In the case of fields of algebraic numbers, the application and development of elements of the Langlands program made it possible to strengthen the Wiles theorem on the Shimura-Taniyama-Weil hypothesis and to prove the Sato-Tate hypothesis.
V. Drinfeld and L. Lafforgue have investigated the case of functional global fields of characteristic p > 0 ( V. Drinfeld for G = GL2 and L. Lafforgue for G = GLr, r is an arbitrary positive integer). They have proved in these cases the Langlands correspondence.
Under the process of these investigations, V. Drinfeld introduced the concept of a F-bundle, or shtuka, which was used by both authors in the proof for functional global fields of characteristic p > 0 of the studied cases of the existence of the Langlands correspondence.
Along with the use of shtukas developed and used by V. Drinfeld and L. Lafforge, other constructions related to approaches to the Langlands program in the functional case were introduced.
G. Anderson has introduced the concept of a t-motive. U. Hartl, his colleagues and students have introduced and have explored the concepts of finite, local and global G-shtukas.
In this review article, we first present results on Langlands program and related representation over algebraic number fields. Then we briefly present approaches by U. Hartl, his colleagues and students to the study of D -shtukas and finite shtukas. These approaches and our discussion relate to the Langlands program as well as to the internal development of the theory of G-shtukas.
Keywords: Langlands correspondence, global field, Drinfeld module, shtuka, finite shtuka, local Anderson-module, cotangent complex, formal group.
Bibliography: 48 titles.
For citation:
N. M. Glazunov, 2020, "On Langlands program, global fields and shtukas" , Chebyshevskii sbornik, vol. 21, no. 3, pp. 68-83.
1. Introduction
This communication is an introduction to the Langlands Program and to (^-)shtukas and finite shtukas (over algebraic curves) over function fields. The Langlands correspondence over number fields in its full generality is facing with problems fl, 2, 3, 4, 5, 6, 7]. So results from Galois theory, algebraic number theory and function fields can help understand it.
1.1. Elements of algebraic number theory and field theory.
The questions what is a Galois group of a given algebraic closure of the number field or the local field, embedding problems of fields and extensions of class field theory belong to fundumental questions of Galois theory and class field theory. A.V. Yakovlev, S.V. Vostokov, B.B. Lur'e works spans many areas of Galois theory, fields theory and class field theory. The results obtained indicate that these questions connect with module theory, homological algebra and with other topics of algebra and number theorv[8, 9, 10, 11, 12, 14, 15]. The development and applications of these theories are discribed in papers by I.R. Shafarevich[4] and by F.N. Parshin [5] (and in references therein). For further details we refer the reader to papers themselves. By the lack of author's competence we discuss here very shortly only connection of local fields with formal modules.
1.2. The Hensel-Shafarevich canonical basis in complete discrete valuation fields.
Vostokov has constructed a canonical Hensel-Shafarevich basis in Zp-module of principle units for complete discrete valuation field with an arbitrary residue field [11]. Vostokov and Klimovitski in paper [13] give construction of primary elements in formal modulei. Ikonnikova, Shaverdova [16] and Ikonnikova [17] use these results under construction, respectively, the Shafarevich basis in higher-dimensional local fields and under proving two theorems on the canonical basis in Lubin-Tate formal modules in the case of local field with perfect residue field and in the case of imperfect residue field. These canonical bases are obtained by applying a variant of the Artin-Hasse function.
1.3. LG to reductive group G
Here we follow to [1, 3, 27, 28, 29]. At first recall that linear algebraic groups found important applications in the Langlands program. Namely, for a connected reductive group G over a global field K, the Langlands correspondence relates automorphic forms on G and global Langlands parameters, i.e. conjugacv classes of homomorphisms from the Galois group Qal(K/K) to the dual Langlands group G(Qp). Let K be an algebraic closure of K and Ks be the separable closure of K in K.
Definition 1. Let, G be the connected reductive algebraic group over K. The root datum, of G is a quadruple (X*(T), A,X*(T), A") where X* is the lattice of characters of the maximal torus T, X* is the dual lattice, given by the 1-parameter subgroups, A is the set of roots, Av is the corresponding set, of coroots.
The dual Langlands group G is a complex reductive group that has the dual root data: (X*(T), Av,X*(T), A). Here any maximal torus T of G is isomorphic to the complex dual torus X *(T) ® C* = Hom(X*(T), C*) of any maximal torus Tin G. Let rQ = Qal(Q/Q).
Given G, the Langlands L-group of G is defined as semidirect product
lG = G x rQ.
In the case of fields of algebraic numbers, the application and development of elements of the Langlands program made it possible to strengthen the Wiles theorem on the Shimura-Taniyama-Weil hypothesis and to prove the Sato-Tate hypothesis. Langlands reciprocity for GLn over non-archimedean local fields of characteristic zero is given by Harris-Taylor [20].
1.4. Langlands correspondence over functional global fields of characteristic
p > 0
V. Drinfeld [6] and L. Lafforgue [7] have investigated the case of functional global fields of characteristic p > 0 ( V. Drinfeld for G = GL2 and L. Lafforgue for G = GLr, r is an arbitrary positive integer). They have proved in these cases the Langlands correspondence.
In the process of these studies, V. Drinfeld introduced the concept of a F -bundle, or shtuka, which was used by both authors in the proof for functional global fields of characteristic p > 0 of the studied cases of the existence of the Langlands correspondence [19].
Along with the use of shtukas developed and used by V. Drinfeld and L. Lafforge, other constructions related to approaches to the Langlands program in the functional case were introduced.
G. Anderson has introduced the concept of a ¿-motive [23]. U. Hartl, his colleagues and postdoc students have introduced and have explored the concepts of finite, local and global G -shtukas[33, 35, 34, 36, 38, 39].
In this review, we first present results on Langlands program and related representation over algebraic number fields. Then we briefly present approaches by U. Hartl, his colleagues and students to the study of G -shtukas. These approaches and our discussion relate to the Langlands program as well as to the internal development of the theory of G-shtukas. Some results on commutative formal groups and commutative formal schemes can be found in [46, 47, 48] and in references therein. The content of the paper is as follows: Introduction.
1. Some results of the implementation of the Langlands program for fields of algebraic numbers and their localizations.
2. Elliptic modules and Drinfeld shtukas.
3. Finite G -shtukas.
The author is grateful to an anonymous reviewer for comments and advice, Dr. Zhivuan Ding for his remark, N.M. Dobrovolskv for help and support in the process of preparing the article for publication.
2. Some results on Langlands program over algebraic number fields and their localizations
Langlands conjectured that some symmetric power L-functions extend to an entire function and coincide with certain automorphic L-functions.
2.1. Abelian extensions of number fields
In the case of algebraic number fields Langlands conjecture (Langlands correspondence) is the global class field theory:
Representations of the abelian Galois group Gal(Kab/K) = characters of the Galois group Gal(K ab/K)
correspond to
automorphic forms on GL1 that are characters of the class group of ideles. Galois group Gal(Kab/K) is the profinite completion of the group A*(K)/K* where A(K) denotes the adele ring of K. If K is the local field, then Galois group Gal(Kab/K) is canonicallv isomorphic to the profinite completion of K*.
2.2. I - adic representations and Tate modules
Let K be a field and K its separate closure, En = {P £ E(K)\nP = 0} the group of points of elliptic curve E(K) order dividing n. When charK does not divide n then En is a free Z/nZ -module of rank 2.
Let I be prime, I = charK. The projective limit Ti(E) of the projective system of modules E\m is free Z|-adic Tate module of rank 2.
Let Vi(E) = Ti(E) Qi. Galois group Gal(K/K) acts on all Eim, so there is the natural continuous representation (¿-adic representation)
pE,i : Gal(K/K) ^ Aut Ti(E) C Aut Vi(E).
Vi (E) is the first homology group that is dual to the first cohomologv group of ¿-adic cohomologv of elliptic curve E and ^obenius F acts on the homology and dually on cohomologv. The characteristic polynomial P(T) of the frobenius not depends on the prime number I.
2.3. Zeta functions and parabolic forms
Let (in P. Deligne notations) X be a scheme of finite type over Z, \X\ the set of its closed points, and for each x £ \X\ let N(x) ^e the ^^^^ct of points of the residue field k(x) of X at x. The Hasse-Weil zeta-function of X is, by definition
a (s)= n (! - N (x)-')-1.
x£\X |
In the case when X is defined over finite field Fg, put qx = N(x), deg(x) = [k(x) : Fg], so qx = qdes(x). Put t = q-s. Then
Z(X,t) = ^ (1 - tde9(x))-1. xe\x\
The Hasse-Weil zeta function of E over Q ^rn of numerators of (e(s) by points of
bad reduction of E) is defined over all primes p:
L(E(Q), s) = n(1 - apP-s + e(p)p1-2s)-1, p
here e(p) = 1 if E has ^^^d reduction at p, md e(p) = 0 otherwise. Put T = p-s. For points of good reduction we have
P(T) = 1 - apT + pT2 = (1 - aT)(1 - pT).
For symmetric power L-functions (functions L(s; E; Symn), n > 0; see below) we have to put
n
Pp(T ) = n(1 - alPn-lT). i=0
For GL2(R), let C ^e its center, 0(2) the orthogonal group.
Upper half complex plane has the representation: H2 = GL2(R)/0(2)C. So it is the
homogeneous space of the group GL2(R).
A cusp (parabolic) form of weight k ^ 1 and level N ^ 1 is a holomorphic function f on the H2
a) For all matrices
g = ^ ac ,a,b,c,d e Z,a = 1(N), d = 1(N), c = 0(N)
and for all 2 e H2 we have
f (gz) = f ((az + b)/(cz + d)) = (cz + d)k f (z)
(automorphic condition).
b)
1/(z)l2(Imz)k
H2
Mellin transform L(f, s) of ^^e parabolic form f coincides with Artin L-series of the representation pf.
The space Mn(N) of cusp forms of weight k and level N is a, finite dimensional complex vector space. If f e Mn(N), then it has expansion
f (z) = r-n(f )exp(2mnz)
n=l
and L-function is defined bv
L(f,s) = ^ cn(f )/ns.
n=1
2.4. Modularity results
The compact Riemann surface T\H2 is called the modular curve associated to the subgroup of finite index r of GL2 (Z) and is denoted bv X(r). If the modular curve is elliptic it is called the elliptic modular curve.
The modularity theorem states that any elliptic curve over Q can be obtained via a rational map with integer coefficients from the elliptic modular curve.
By the Hasse-Weil conjecture (a cusp form of weight two and level N is an eigenform (an eigenfunction of all Hecke operators)). The conjecture follows from the modularity theorem. Recall the main (and more stronger than in Wiles [21] and in Wiles-Taylor [22] papers) result by C. Breuil, B. Conrad, F. Diamond, R. Taylor [24].
Theorem 1. fTaniyama-Shimura-Weil conjecture - Wiles Theorem.^ For every elliptic curve E over Q there exists f, a cusp form of weight 2 for a subgroup r0(N), such that L(f,s) = L(E(Q),s).
Here r0(N) is the modular group
ro(N ac bd^j,a,b,c,d G Z, c = 0 (mod N), det ^ ^ .
Recall that for projective closure E of the elliptic curve E we have
E(FP) = 1 - ap + p.
By H. Hasse
ap = 2^pcos (pp.
Conjecture 1. (Sato-Tate conjecture) Let E be an elliptic curve without complex multiplication. Sato have computed and Tate gave theoretical evidence that angles <pp in the case are equidistributed in [0, k] with the Sato-Tate density measure 2 sin2 tp.
We have two theorems from Serre [18] which give the theoretical explanation in terms of Galois representations. Here we recall the corollerv of the theorems.
corollary 1. (Serre [18]) The elements are equidistributed for the v normalized Haar measure of G if and only if c = 0 for every X irreducible character of G, i. e., if and only if the L-functions relative to the non trivial irreducible characters of G are holomorphic and non zero at s = 1.
The current state of Sato-Tate conjecture is now Clozel-Harris-Shepherd-Barron-Taylor Theorem [25, 26].
Theorem 2. (Clozel, Harris, Shepherd-Barron, Taylor). Suppose E is an elliptic curve over Q with non-integral j invariant. Then for all n > 0; L(s; E; Symn) extends to a meromorphic function which is holomorphic and non-vanishing for Re(s) ^ 1 + n/2.
These conditions and statements are sufficient to prove the Sato-Tate conjecture.
Under the prove of the Sato-Tate conjecture the Tanivama-Shimura-Weil conjecture oriented methods of A. WTiles and R. Taylor are used.
Recall also that the proof of Langlands reciprocity for GLn over non-archimedean local fields of characteristic zero is given by Harris-Taylor [20].
3. Elliptic modules and Drinfeld shtukas.
Let
Fg be the algebraic closure of Fg,
C be a smooth projective geometrically irreducible curve over Fg,
K be the function field Fq(C) of C,
v be a close point of C,
A be the ring of functions regular on C - v,
Ku be the complation of K at v with valuation ring Ov,
C be the complation of the algebraic closure of Ku.
At first recall some known facts about algebraic curves over finite fields. We will identify the set \C\ of closed points of C with C(Fg) = HomFq(Spec Fg, C). Let k(v) be the residue field of v. Then the degree of v is equal of ^te number of elements [k(u) : Fg]. Below in this section we follow to [30, 19, 31].
3.1. Elliptic modules
Lemma 1. Let k be a field of characteristic p > 0 and Iet R be a k-commutative ring with unit (there exists a morphism k ^ R). The additive scheme Ga over R is represented by the polynomial ring R[X] with structural morphism a : R[X] ^ R[X] R[X], given by a(X) = X 0 1 + 1 ® X. A morphism p : Ga ^ Ga of additive schemes over R is defined by an additive polynomial. If ^ is another such morphism, then p o ^ = )). So the set of (endo)morphisms of additive scheme
has the structure of a ring.
Example 1. Let a £ R[X], pa = 0. Then the morphism p(T) = aTp", (n ^ 0) is additive. Any additive morphism p(T) in characteristic p has the form p(T) = a0T + a1Tp + ■ ■ ■ + anTp .
Proposition 1. Let, k be a field of characteristic p > 0. Putra = apT. There is an isomorphism, between Endk(Ga) and the ring of noncommutative polynomials k{r}.
For any p(T) = a0T+aiTp +-----+anTp" e Endk(Ga) and any p(r) = a0+air +-----+anTn e k{r}
Lubin morphisms [32] c^d c are defined:
co(<p(T)) = ao,c(ip(r)) = ao.
Respectively we define
deg(<p(T)) = pn,d(p(r)) = n.
Proposition 2. Any ring morphism A ^ Endk(Ga) is either injective or has image contained in the constants k C k{r}.
Sketch of the proof. k{r} is a domain. Endk(G) is isomorphic to k{r}. A is a ring with divisor theory D and fa any prime divisor p e D the residue ring A/p is a field. From these statements the proposition follows.
Assume now that k is an A-algebra, i.e. there is a morphism i : A ^ k.
Definition 2. An elliptic module over k (of rank r = 2) is an injective ring h,om,om,orph,ism,
p : A ^ Endk(Ga) a ^ pa,
such that for all a e A we have
d(p(r)) = 2 ■ deg(a), c(p(r)) = i(a).
Example 2. Let k = Fg(T), A = Fg[P1 - v\= Fg[T]. Let i(T) = T2 + 1. In this case an elliptic module p is given by
p = T2 + 1 + Ci ■ T + C2 ■ T2, Ci, C2 e k, C2 = 0.
Remark 1. By the same way it is possible to define a Drinfeld module (over a field) for any natural r.
Now consider the case of Drinfeld modules over a base scheme. Let S be an A-scheme, C a line bundle over S, i* : S ^ Spec A be an A scheme morphism dual to the ring homomorphism i : A
Definition 3. (Drinfeld module over a base scheme) A Drinfeld module over k of rank r is an ring homomorphism,
p : A ^ Ends(C)
a ^ Pa,
such that for all a e A we have
1) locally, as a polynomial in t, pa has the degree
d(p(r)) = r ■ deg(a),
2) a unit, as its leading coefficient an and
c(p(r)) = i(a).
3.2. Drinfeld shtukas.
In notations of previous subsection let x e k, a e A, ) be a Drinfeld module of rank r. Put L = k{r}, f (t) e L, k[A] = k ®Fq A, degT f (r) the degree in r of f (r).
Lemma 2. Define the action of k[A\ on L by the formula:
X ® a ■ f (t) = x ■ f (pa(t). Then L is a free k[A\-module of rank r.
Remark 2. Let, Es = {f(r) e LldegTf(r) 4 s}, E = ®f=0Es, E[1\ = ®f=0Es+i. E,E[1] are graded modules over the graded ring and give rise to locally free sheaves F, £ of rank r over C.
Put Cs = C xWq S, aq = idc ® Frobq,s : Cs
Definition 4. A (right) V-shtuka (F-sheaf [19]) of rank r over an Fq-scheme S is a diagram (F % S?2 (idc ® Evobq,s)*F), such that coker c1 is supported on the graph ra of a morphism a : S % C and it is a line bundle on support, coker c2 is supported on the graph, r@ of a morphism P : S % C and it is a line bundle on support.
If ra n rp = 0 it is possible to give the next definition of V-shtuka [34, 37].
Definition 5. A global shtuka of rank r with two legs over an Fq-scheme S is a tuple M = (M, (c1,c2),tN) consisting of 1) a locally free sheaf N of rank, r on Cs; 2) Fq-m,orph,ism,s Ci : S % C (i = 1,2), called the legs of N; 3) an isomorphism tn : a**Nlcs-(r ur^) — Nlcs-(rciurC2) outside the graphs rCi of a, rci n rc2 = 0.
Definition 6. A global shtuka over S is a V-shtuka if tn satisfies tn(&**N) C M on Cs — rC2 with cokernel locally free of rank 1 as Os-module, and t-l(M) C o*N on Cs — rci with, cokernel locally free of rank 1 as Os-module.
4. Finite G-shtukas.
We follow to [19, 33, 35, 36, 37]. We start with very short indication on the general framework of the section. In connection with Drinfeld's constructions of elliptic modules Anderson [23] has introduced abelian t-modules and the dual notion of t-motives. Beside with mentioned papers these are the descent theory by A. Grothendieck [40], cotangent complexes by Illusie [44], by S. Lichtenbaum and M. Schlessinger [41], by Messing [42] and by Abrashkin [43]. In this framework to any morphism f : A % B of commutative ring objects in a topos is associated a cotangent complex L(b/a) and to any morphism of commutative ring objects in a topos of finite and locally free 5pec(A)-group schemes G is associated a cotangent complex L(G/spec(A)) as has presented in books by Illusie [44].
4.1. Finite shtukas and formal groups
Let S be a scheme over Spec Fg.
Definition 7. A finite Fg-sMwia over S is a pair M = (M,Fm) consisting of a locally free Os-module M on S of finite rank and an Os-module homomorphism Fm : oM % M.
Author [36] investigates relation between finite shtukas and strict finite flat commutative group schemes and relation between divisible local Anderson modules and formal Lie groups. The
cotangent complexes as in papers by S. Lichtenbaum and M. Schlessinger [41], by W. Messing [42], by V. Abrashkin [43] are defined and are proved that they are homotopicallv equivalent.
Then the deformations of afline group schemes follow to the mentioned paper of Abrashkin are investigated and strict finite 0-module schemes are defined. Next step of the research is devoted to relation between finite shtukas by V. Drinfeld [19] and strict finite flat commutative group schemes. The comparison between cotangent complex and Frobenius map of finite Fp-shtukas is given.
4.2. Local shtukas and local Anderson modules
Recall some notions and notations. An ideal I in a commutative ring A is locally nilpotent at a prime ideal q if the localization Ie is a nilpotent ideal in Ae. In the framework of smooth projective geometrically irreducible curves C over Fg let NUpav denote the category of A^-schemes on which the uniformizer £ of Au is locally nilpotent. Here Au ~ F v [[£]] is the completion of the local ring Oc,u at a closed point v £ C.
Let NilpFq[£]] be the category of Fg[[£]]-schemes on which £ is locally nilpotent. Let S £ NilpFq Let M be a sheaf of Os[[z]]-modules on S and let a**M = M 0Os\[z\],a* Os[[z]], M[] = M0Os[W\ 0s [[z]][ ^ 'q
Definition 8. A local shtuka of height r over S is a pair M = (M, Fm) consisting of a locally free sheaf M of Os[[z]]-modules of rank r, and an isomorphism Fm : &*qM[] ~ M[].
The next lemma is proved [37].
Lemma 3. Let R be an Fgin which £ is nilpotent. Then the sequence of R[[z]]-modules
0 ^ R[[z]] ^ R[[z]] ^ R ^ 0 z - £
is exact. In particular E[[z]] C ^[[^]][j—e]•
In the conditions of the lemma authors [37] give the next
Definition 9. A z-divisible local Anderson module over R is a sheaf ofFq[[z]]-modules G on the big fppf-site of Spec R such that
(a) G is z-torsion, that is G = G[zn], where G[zn] = ker(zn : G ^ G),
(b) G is z-divisible, that is z : G ^ G is an epimorphism,
(c) For every n the Fq-module G[zn] is representable by a finite locally free strict Fq-module scheme over R in the sense of Faltings ([4-5, 37]), and
(d) locally on Spec R there exists an integer d £ Zsuch that (z - £)d = 0 on ug where ug = Jim uG[zn\ and UG\zn\ = ^q^1G[zn\/sPec R for the unit section e of G[zn] over R.
z-divisible local Anderson modules by Hartl [33] with improvements in [37] and local shtukas are investigated. The equivalence between the category of effective local shtukas over S and the category of z-divisible local Anderson modules over S is treated by the authors [36, 37]. The theorem about canonical Fp[[£]] -isomorphism of z-adic Tate-module of z -divisible local Anderson module G of rank r over S and Tate module of local shtuka over S associated to G is given. The main result of [36] is the following (section 2.5) interesting result: it is possible to associate a formal Lie group to any ¿-divisible local Anderson module over S in the case when £ is locally nilpotent on S. We note that related with [36] and in some cases more general results have presented in the paper by U. Hartl, E. Viehmann [35].
СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
1. R. P. Langlands. On the notion of an automorphic representation. Automorphic Forms, Representations, and L-Functions // Proc. Svmp. Pure Math., vol. 33, Part I, AMS, Providence, 1979, 203-207.
2. R. P. Langlands. Base change for GL(2) // Annals of Math. Studies, vol. 96, Princeton Univ. Press, Princeton, 1980.
3. H. Jacquet, R. P. Langlands. Automorphic Forms on GL(2) // Lecture Notes in Mathematics, vol. 114, Springer-Verlag, Berlin, 1970.
4. I.R. Shafarevich. Abelian and nonabelian mathematics (in Russian) // Sochineniva (Works), v.3, part 1, Moscow, Prima-B, 1996, 397-415.
5. A.N. Parshin. Questions and remarks to the Langlands program, // Uspekhi Matem. Nauk, 67, n 3, 2012, 115-146.
6. V. Drinfeld. Langlands' conjecture for GL(2) over functional fields // Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, 565-574.
7. L. Lafforgue. Chtoucas de Drinfeld, formule des traces d'Arthur-Selberg et correspondance de Langlands // Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), Higher Ed. Press, Beijing, 2002, 383-400.
8. I.R. Shafarevich. On embedding problem for fields // Math. USSR-Izv., vol. 18, 1954, 918-924.
9. A.V. Yakovlev. The embedding problem for number fields // Math. USSR-Izv., vol. 31, no. 2, 1967, 211-224.
10. A.V. Yakovlev. Galois group of the algebraic closure of local fields // Math. USSR-Izv., vol. 32, no. 6, 1968, 1283-1322.
11. S.V. Vostokov. Canonical Hensel-Shafarevich basis in complete discrete valuation fields // Zap. Nauchn. Semin. POMI, 394, 2011, 174-193.
12. S.V. Vostokov. Explicit construction of class field theory for a multidimensional local field // Math. USSR-Izv., vol. 49, no. 2, 1985, 283-308.
13. S.V. Vostokov, I.L. Klimovitskii. Primerv elements in formal modules // Mathematics and Informatics, 2, Steklov Math. Inst., RAS, Moscow, 2013, 153-163.
14. В. B. Lur'e. On embedding problem with kernel without center // Math. USSR-Izv., vol. 28, 1964, 1135-1138.
15. В. B. Lur'e. Universally solvable embedding problems // Proc. Steklov Inst. Math., 183, 1991, 141-147.
16. E.V. Ikonnikova, E.V. Shaverdova. The Shafarevich basis in higher-dimensional local field // Zap. Nauchn. Semin. POMI, 413, 2013, 115-133.
17. E.V. Ikonnikova. The Hensel-Shafarevich canonical basis in Lubin-Tate formal modules //J. Math. Sci., New York 219, No. 3, 2016, 462-472.
18. J.-P. Serre. Abelian ¿-Adic Representations and Elliptic Curves // Addison-Wesley Publishing Company, NY, 1989.
19. V. Drinfeld. Moduli varieties of F-sheaves // Func. Anal, and Applications, vol. 21, 1987, 107-122.
20. M. Harris, R. Taylor. The geometry and cohomologv of some simple Shimura varieties // Ann. Math. Stud. 151, Princeton University Press, Princeton, 2001.
21. A. WTiles. Modular elliptic curves and Fermat's last theorem //Ann. of Math. (2) 141, no. 3, 1995, 443-551.
22. R. Taylor, A. WTiles. Ring-theoretic properties of certain Hecke algebras // Ann. of Math. (2) 141, no. 3, 1995, 553-572.
23. G. Anderson. t-Motives // Duke math. J., vol. 53, 1986, 457-502.
24. C. Breuil, B. Conrad, F. Diamond, R. Taylor. On the modularity of elliptic curves over Q: wild 3-adic exercises //J. Amer. Math. Soc. 14, no. 4, 2001, 843-939.
25. L. Clozel, M. Harris, R. Taylor. Automorphv for some 1-adic lifts of automorphic mod 1 Galois representations // Pub. Math. I.H.E.S. 108*2008, 1-181.
26. M. Harris, N. Shepherd-Barron, R. Taylor. A family of Calabi-Yau varieties and potential automorphv // Ann. of Math. (2) vol. 171, no. 2, 2010, 779-813.
27. Arthur J. The principle of functorialitv // Bulletin of the Amer. Math. Soc., Vol. 40, no. 1, 39-53, 2003.
28. A. Borel. Automorphic L-functios // Proc. Svmp. Pure Math., vol. 33, Part 2, 1979, 27-61.
29. Tate J. Number theoretic background, Proc. Svmp. Pure Math., vol. 33, Part 2, pp. 3-26, 1979.
30. V. Drinfeld. Elliptic modules. Mat. Sbornik, vol. 94, no. 4, pp. 594-627, 1974.
31. P. Deligne, D. Husemoller. Survey of Drinfeld's modules // Contemporary Math., vol. 67, 25-91, 1987.
32. J. Lubin. One-parameter formal Lie groups over p-adic integer rings // Ann. of Math., vol. 80, 1964, 464-484.
33. Urs Hartl. Number Fields and Function fields - Two Parallel Worlds, Papers from the 4th Conference held on Texel Island, April 2004, // Progress in Math. 239, Birkhauser-Verlag, Basel, 2005, 167-222.
34. Urs Hartl, Arasteh Rad. Local P-shtukas and their relation to global G-shtukas // Miinster J. Math. 7, No. 2, 2014, 623-670.
35. Urs Hartl, E. Viehmann. // J. reine angew. Math. (Crelle) 656, 2011, 87-129.
36. R. Singh. Local shtukas and divisible local Anderson-modules // Univ. Miinster, Fachbereich Mathematik und Informatik (Diss.). 72 p. 2012.
37. Urs Hartl, R. Singh. Local Shtukas and Divisible Local Anderson Modules, // Canadian J. of Math., vol. 71, no. 5, 2019, 1163-1207.
38. Arasteh Rad, Uniformizing the moduli stacks of global ©-shtukas // Univ. Münster, Fachbereich Mathematik und Informatik, (Diss.). 85 p. 2012.
39. A. Weiß. Foliations in moduli spaces of bounded global G-shtukas // Univ. Münster, Fachbereich Mathematik und Informatik, (Diss.). 97 p. 2017.
40. A. Grothendieck. Catégories fibrèes et descente, Exposé VI in Revêtements étales et groupe fondemental (SGA 1), Troisième édition, corrigé // Institut des Hautes 'Etudes Scientifiques, Paris, 1963.
41. S. Lichtenbaum, M. Schlessinger. The cotangent complex of morphisms // Transactions of the American Math, society, 128, 1967, 41-70.
42. WT. Messing. The Cristals Associated to Barsotti-Tate Groups // LNM 264, Springer-Verlag, Berlin etc.1973.
43. V. Abrashkin. Compositio Mathematika, 142:4, 2006, 867-888.
44. L. Illusie. Complex cotangent et deformations. I, II // LNM, Vol.239, Vol. 283, Springer Verlag, Berlin-NY, 1971, 1972.
45. G. Faltings. Group schemes with strict 0-action // Mose. Math. J. 2, no. 2, 2002, 249-279.
46. N. Glazunov. Quadratic forms, algebraic groups and number theory // Chebvshevskii Sbornik, vol.16, no. 4, 2015, 77-89.
47. N. Glazunov. Extremal forms and rigidity in arithmetic geometry and in dynamics // Chebvshevskii Sbornik, vol.16, no. 3, 2015, 124-146.
48. N. Glazunov. Duality in abelian varieties and formal groups over local fields. I. Chebvshevskii Sbornik, vol. 19, no.l, 2018, 44-56.
REFERENCES
1. Langlands R. P. On the notion of an automorphic representation, Automorphic Forms, Representations, and L-Functions, Proc. Svmp. Pure Math., vol. 33, Part I, American Mathematical Society, Providence, pp. 203-207, 1979.
2. Langlands R. P., Base change for GL(2), Annals of Math. Studies, vol. 96, Princeton Univ. Press, Princeton, 1980.
3. Jacquet, H., and R. P. Langlands, Automorphic Forms on GL(2), Lecture Notes in Mathematics, vol. 114, Springer-Verlag, Berlin, 1970.
4. Shafarevich I.R. Abelian and nonabelian mathematics (in Russian), Sochineniva (Works), v.3, part 1, Moscow, Prima-B, 397-415, 1996.
5. Parshin A.N. Questions and remarks to the Langlands program, Uspekhi Matem. Nauk, 67, n 3, 115-146, 2012.
6. Drinfeld V. Langlands' conjecture for GL(2) over functional fields. Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 565-574, Acad. Sei. Fennica, Helsinki, 1980.
7. Lafforgue L. Chtoucas de Drinfeld, formule des traces d'Arthur-Selberg et correspondance de Langlands. Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), pp. 383-400, Higher Ed. Press, Beijing, 2002.
8. Shafarevich I.R. On embedding problem for fields. Math. USSR-Izv., vol. 18, P. 918-924, 1954.
9. Yakovlev A.V. The embedding problem for number fields. Math. USSR-Izv., vol. 31, no. 2, P. 211-224, 1967.
10. Yakovlev A.V. Galois group of the algebraic closure of local fields. Math. USSR-Izv., vol. 32, no. 6, P. 1283-1322, 1968.
11. Vostokov S.V. Canonical Hensel-Shafarevich basis in complete discrete valuation fields, Zap. Nauchn. Semin. POMI, 394, 174-193, 2011.
12. Vostokov S.V. Explicit construction of class field theory for a multidimensional local field Math. USSR-Izv., vol. 49, no. 2, 283-308, 1985.
13. Vostokov S.V. Klimovitskii I.L. Primerv elements in formal modules, Mathematics and Informatics, 2, Steklov Math. Inst., RAS, Moscow, 153-163, 2013.
14. B. B. Lur'e. On embedding problem with kernel without center Math. USSR-Izv., vol. 28, 1135-1138,1964.
15. B. B. Lur'e. Universally solvable embedding problems, Proc. Steklov Inst. Math., 183, 141-147, 1991.
16. Ikonnikova E.V., Shaverdova E.V. The Shafarevich basis in higher-dimensional local field, Zap. Nauchn. Semin. POMI, 413, 115-133, 2013.
17. Ikonnikova E.V. The Hensel-Shafarevich canonical basis in Lubin-Tate formal modules, J. Math. Sci., New York 219, No. 3, 162 72. 2016.
18. Serre J.-P. Abelian ¿-Adic Representations and Elliptic Curves, Addison-Wesley Publishing Company, NY, 1989.
19. Drinfeld V. Moduli varieties of F-sheaves. Func. Anal, and Applications, vol. 21, pp. 107-122, 1987.
20. Harris M., Taylor R. The geometry and cohomologv of some simple Shimura varieties, Ann. Math. Stud. 151, Princeton University Press, Princeton, 2001.
21. WTiles, A., Modular elliptic curves and Fermat's last theorem. Ann. of Math. (2) 141, no. 3, 443-551, 1995.
22. Taylor, R.; WTiles, A., Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141, no. 3, 553-572, 1995.
23. Anderson G. t-Motives. Duke math. J., vol. 53, pp. 457-502, 1986.
24. Breuil, C., Conrad, B., Diamond F., Taylor R., On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Amer. Math. Soc. 14, no. 4, 843-939, 2001.
25. Clozel L., Harris M., Taylor, R., Automorphv for some 1-adic lifts of automorphic mod 1 Galois representations, Pub. Math. I.H.E.S. 108, 1-181, 2008.
26. Harris M., Shepherd-Barron N., Taylor R. A family of Calabi-Yau varieties and potential automorphv, Ann. of Math. (2) vol. 171, no. 2, 779-813, 2010.
27. Arthur J. The principle of functorialitv, Bulletin of the American Math, society, Vol. 40, no. 1, pp. 39-53, 2003.
28. Borel A. Automorphic L-functios, Proc. Svmp. Pure Math., vol. 33, Part 2, pp. 27-61, 1979.
29. Tate J. Number theoretic background, Proc. Svmp. Pure Math., vol. 33, Part 2, pp. 3-26, 1979.
30. Drinfeld V. Elliptic modules. Mat. Sbornik, vol. 94, no. 4, pp. 594-627, 1974.
31. Deligne P., Husemoller D., Survey of Drinfeld's modules, Contemporary Math., vol. 67, pp. 25-91, 1987.
32. Lubin J. One-parameter formal Lie groups over p-adic integer rings, Ann. of Math., vol. 80, pp. 464-484, 1964.
33. Urs Hartl. Number Fields and Function fields - Two Parallel Worlds, Papers from the 4th Conference held on Texel Island, April 2004, Progress in Math. 239, Birkhauser-Verlag, Basel, pp. 167-222, 2005.
34. Hartl U., Arasteh Rad. Local P-shtukas and their relation to global G-shtukas. Münster J. Math. 7, No. 2, 623-670, 2014.
35. Hartl U., Viehmann E. J. reine angew. Math. (Grelle) 656, 87-129, 2011.
36. Singh R., Local shtukas and divisible local Anderson-modules, Univ. Münster, Fachbereich Mathematik und Informatik (Diss.). 72 p. 2012.
37. Hartl U., Singh R., Local Shtukas and Divisible Local Anderson Modules, Canadian J. of Math., vol. 71, no. 5, 1163-1207, 2019.
38. Arasteh Rad, Uniformizing the moduli stacks of global ©-shtukas, Univ. Münster, Fachbereich Mathematik und Informatik, (Diss.). 85 p. 2012.
39. Weiß A, Foliations in moduli spaces of bounded global G-shtukas. Univ. Münster, Fachbereich Mathematik und Informatik, (Diss.). 97 p. 2017.
40. Grothendieck A. Catégories fibrèes et descente, Exposé VI in Revêtements étales et groupe fondemental (SGA 1), Troisième édition, corrigé, Institut des Hautes 'Etudes Scientifiques, Paris, 1963.
41. Lichtenbaum S., Schlessinger M. The cotangent complex of morphisms, Transactions of the American Math, society, 128, pp. 41-70. 1967.
42. Messing W. The Cristals Associated to Barsotti-Tate Groups, LNM 264, Springer-Verlag, Berlin etc.1973.
43. Abrashkin V. Compositio Mathematika, 142:4, pp. 867-888, 2006.
44. Illusie L. Complex cotangent et deformations. I, II, LNM, Vol.239, Vol. 283, Springer Verlag, Berlin-NY, 1971, 1972.
45. Faltings G. Group schemes with strict 0-action, Mose. Math. J. 2, no. 2, 249-279, 2002.
46. Glazunov N., Quadratic forms, algebraic groups and number theory, Chebvshevskii Sbornik, vol.16, no. 4, P.77-89, 2015.
47. Glazunov N., Extremal forms and rigidity in arithmetic geometry and in dynamics, Chebvshevskii Sbornik, vol.16, no. 3, P. 121 116. 2015.
48. Glazunov N., Duality in abelian varieties and formal groups over local fields. I. Chebvshevskii Sbornik, vol. 19, no.l, P. 44-56, 2018.
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