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ЧЕБЫШЕВСКИЙ СБОРНИК Том 22. Выпуск 2.
УДК 51 DOI 10.22405/2226-8383-2021-22-2-90-103
Расширения Любина — Тейта и модуль Карлица над проективной прямой: явная связь
Н. В. Елизаров, С. В. Востоков
Елизаров Никита Вячеславович — магистрант, Санкт-Петербургский государственный университет (г. Санкт-Петербург). e-mail: nikich97@bk.ru
Востоков Сергей Владимирович — доктор физико-математических наук, Санкт-Петербургский государственный университет (г. Санкт-Петербург). e-mail: s.vostokov@spbu.ru
Аннотация
В данной статье рассматриваются различные подходы к построению максимальных абелевых расширений для локальных и глобальных геометрических полей. Теория Любина — Тейта играет ключевую роль в построении максимального Абелева расширения для локальных геометрических полей. В случае глобальных геометрических полей особый интерес представляют модули Дринфельда. В настоящей работе рассматривается самый простой частный случай модулей Дринфельда для проективной прямой, который называется модулем Карлица.
Во введении мы приводим мотивацию и краткую историческую справку по затронутым в работе темам.
В первом и втором разделах мы приводим краткую информацию о модулях Любина-Тейта и модуле Карлица.
В третьем разделе мы приводим два основных результата:
• установлена явная связь между теориями глобальных и локальных полей в геометрическом случае проективной прямой над конечным полем: доказано, что башня расширения модуля Карлица индуцирует башню расширений Любина-Тейта.
• установлена связь между отображениями Артина расширений функционального поля произвольной проективной гладкой неприводимой кривой и расширениями пополнений локальных колец в замкнутых точках этой кривой.
В последнем разделе мы формулируем различные открытые задачи и интересные направления для дальнейших исследований, которые включают обобщение первого результата для произвольной гладкой проективной кривой над конечным полем и рассмотрение модулей Дринфельда более высокого ранга.
Ключевые слова: теория полей классов, теория Любина — Тейта, модуль Карлица, модули Дринфельда, отображение Артина, максимальное абелево расширение, проективная прямая над конечным полем.
Библиография: 11 названий. Для цитирования:
Н. В. Елизаров, С. В. Востоков. Расширения Любина — Тейта и модуль Карлица над проективной прямой: явная связь // Чебышевский сборник, 2021, т. 22, вып. 2, с. 90-103.
CHEBYSHEVSKII SBORNIK Vol. 22. No. 2.
UDC 51 DOI 10.22405/2226-8383-2021-22-2-90-103
Lubin—Tate extensions and Carlitz module over a projective line: an explicit connection
N. V. Elizarov, S. V. Vostokov
Elizarov Nikita Vyacheslavovich — graduate student, Saint Petersburg State University (St.
Petersburg).
e-mail: nikich97@bk.ru
Vostokov Sergei Vladimirovich — doctor of physical and mathematical sciences, Saint Petersburg State University (St. Petersburg). e-mail: s.vostokov@spbu.ru
Abstract
In this article we consider different approaches for constructing maximal abelian extensions for local and global geometric fields. The Lubin-Tate theory plays key role in the maximal abelian extension construction for local geometric fields. In the case of global geometric fields, Drinfeld modules are of particular interest. In this paper we consider the simpliest special case of Drinfeld modules for projective line which is called the Carlitz module.
In the introduction, we provide motivation and a brief historical background on the topics covered in the work.
In the first and second sections we provide brief information about Lubin-Tate modules and Carlitz module.
In the third section we present two main results:
• an explicit connection between the local and global field theory in the geometric case for projective line over finite field: it is proved that the extension tower of Carlitz module induces the tower of the Lubin-Tate extensions.
• a connection between Artin maps of extensions of a function field of an arbitrary projective smooth irreducible curve and extensions of completions of local rings at closed points of this curve.
In the last section we formulate different open problems and interesting directions for further research, which include generalization first result for an arbitrary smooth projective curve over a finite field and consideration Drinfeld modules of higher rank.
Keywords: class field theory, Lubin-Tate theory, Carlitz module, Drinfeld modules, Artin map, maximal abelian extension, projective line over a finite field.
Bibliography: 11 titles. For citation:
N. V. Elizarov, S. V. Vostokov, 2021, "Lubin-Tate extensions and Carlitz module over a projective line: an explicit connection", Chebyshevskii sbornik, vol. 22, no. 2, pp. 90-103.
Introduction
The main motivation of this work is the study of Hilbert's 9th problem, in particular, an attempt to transfer the results of S. V. Vostokov [1] to the case of geometric fields. This problem is of high interest in modern algebraic number theory and has been discussed in numerous scientific papers [2] - [3].
In 1853, the famous Kronecker-Weber theorem was proved for the arithmetic global case. It says that an arbitrary finite abelian extension of the field of rational numbers lies in some cyclotomic extension of Q.
Consider a geometric analogue of this statement. Let Fg(X) be a field of rational functions of the projective line X = P^ over a finite field Fg. Using the theory of Carlitz module, we can build explicitly cyclotomic extensions of Fg(X) and construct the maximum abelian extension of the global field Fg(X).
Teruyoshi Yoshida built the maximum abelian extension and the Artin map for an arbitrary local field of an arbitrary characteristic [6]. In the construction, the theory of formal Lubin-Tate modules was applied.
The first result of this paper is a construction of a connection between theories of building a maximal abelian extension for local and global fields. It is proved that the extension tower of Carlitz module of the global field induces the extension tower of formal Lubin—Tate modules over the completion of the local ring at the closed point of our curve Theorem 1.
The second result is a description of the connection between Artin mappings for an arbitrary projective irreducible smooth curve X and for completions of local rings at its closed points Theorem 2.
Acknowledgement. The research is supported by the Russian Science Foundation under grant N 16-11-10200.
Preliminaries and notation Local fields
Throughout the work all the considered fields have characteristic other than 2.
Let p be a prime integer not equal to 2. q = ps, where s is an arbitrary natural number. Then for Fg we denote a finite field consisting of q elements.
We call a field K with non-Archimedean regular valuation uK local if:
• K is complete with respect to
• residue field with respect to vK is finite.
We denote ring of integers, simple ideal and residue field by OK, pK and tK respectively. If K' is a finite extension of local field K we denote ramification index and the degree of inertia by eK//K and ftt/R respectively.
If eK'/K = 1, then the extension is called finite unramified. If fK/K = 1, then we call such extension as finite completely ramified. Finally, if eK//K is coprime with p, then the extension is called finite weakly ramified, and the ramification itself is called finite tamely ramification.
Now we need to take a closer look on infinite extensions.
A separable extension E/K is said to be unramified (completely ramified) if the extension is obtained by the union of finite unramified (completely ramified) extensions over K. We assume that the ring of integers of an arbitrary separable extension E/K is the complete closure of the ring of integers OK in E. In the case when all our extensions are finite, this definition coincides with the
classical one. We also call separable extension E/K a finitely-ramified if E/K is a finite extension of some unramified extension.
It is well-known fact that for every positive integer n and arbitrary local field K there is a unique finite unramified extension Kn such that [K. : K] = n. Moreover, Gal(K^/K) = Gal(tKn/t) = (Z/nZ)x. We denote the Frobenius automorphism, the generator of Gal(K^/K), by A.
Maximal unramified extension of the local field K is IJ K. and we denote it by Kur. It is
raGN
unramified by the very definition. What is more, Gal(KMr/k) is a protective limit of Gal(K^/K) :
Gal(KOT/K) = limGal(K„/K) = lim(Z/nZ)x = Z.
A Frobenius automorphism is the unique automorphism A in Gal(KMr/K) with the property A(^) = xq(mod pKur) for all x £ 0Kur. The Frobenius element maps to identity under the isomorphism lim(Z/nZ)x = Z. Restriction of this automorphism on an arbitrary finite unramified field Kn gives
us the corresponding Frobenius automorphism in the finite extension. It explains the name. Proposition 1. Let E/K be a finitely ramified separable extension.
• Oe is a discrete valuation ring. Moreover, the valuation obtained from this ring is a continuation of the valuation vK. According to this valuation, we can take the completion E. We denote it by E.
• If E'/E is a finite separable extension, then E' ■ E = E'.
• E n Ksep = E. In particular, for finitely ramified extensions E, E' over K if E = E', then E = E'.
We call the field L a complete extension of K if L is the completion of some finitely ramified separable extension of K. L is a complete unramified extension if it is the completion of an unramified separable extension of K.
For convenience we denote the completion of Kur with repect to uK by F. If L/K is a complete unramified extension of K, and L' is a completely ramified extension of L, then we say that this is a Galois extension if any automorphism ^ £ Aut(L/K) extends to [V : L] different automorphisms in Aut(L'/K). We call the Weil group of such Galois extension a W(L'/K) := {a £ Aut(L'/K) | a\L £ AZ}. If L = F, then we introduce the map:
v : W(L'/K) ^ Z;
Formal group laws
By a formal group law over an arbitrary ring R we mean a series F(X,Y) £ R[[X, Y]] which satisfies:
• F(X, Y) = X + Y(mod deg 2);
• F(X, F(Y, Z)) = F(F(X, Y), Z) (associativity);
• F(X,Y) = F(Y,X) (commutativity).
Lubin—Tate extensions and maximal abelian extensions
Definition 1. We call the polynomial f e O#[Xj Lubin-Tate polynomial if it satisfies two properties:
1. f (X) = n ■ X (mod deg 2);
2. f (X) = X* (mod pF),
where by n we mean the arbitrary unifomizer of local field K. Let us also define the action of A
on an arbitrary formal group law F(X,Y) e O$[[X,Yjj. If F(X,Y) = E aijXiY^ then Ff :=
ij>l
E A(a,ij)XiY3.
Then for arbitrary Lubin-Tate polynomial f it is known [6, Lemma 3.4] that there exists a unique formal group law F(X, Y) e O#[[X, Yjj:
f o F = Ff o f.
We call such a law a formal group law which corresponds to f and denote it by Ff.
Lubin-Tate polynomials are very important in the explicit construction of the maximal abelian extension and Artin map for an arbitrary local field. In the general situation Lubin-Tate polynomials are generalized to Lubin-Tate series with coefficients not necessary in Of but in Ol, where L can be an arbitrary complete unramified extension of K. However, in the considered here cases the series belongs to OK[Xj. The next proposition partly explains it.
Proposition 2. If f e Ok[Xj, then Ff e OM[[X, Yjj and, therefore, Ff = Ff.
Proof. The idea of proof is taken from [6, Lemma 3.4]. It is enough to construct such a sequence of polynomials {Fm} satisfying two properties:
1. deg Fm ^ m;
2. f o Fm = Ff o f (mod deg (m + 1)).
However, due to our needs we only prove that Fm e OK[X, Yj by induction.
The base case is simple, because F\ = X + Y and required property is true. To understand inductive step we define Hm+\ = Fm+\ — Fm and Gm+\ as f o Fm — Ff o f. To find Fm+\ it is enough to find Hm+\.
From [6, Lemma 3.4] we obtain that if k ■ fy is the coefficient at XiYJ in Gm+\, then the coefficient a^ at the same monomial in Hm+\ satisfies the equality:
<x 1—1
= —Pij — En A>m) ■ A'). 1=1 i=0
According to the induction hypothesis, A(^iJ-) = fi^. We note that since k e OK, then A(^) = n. Thus, we get A(aiJ-) = a^. □
Now we briefly introduce Lubin-Tate modules. As we already mentioned, we work mostly with polynomials from OK[Xj. The technical details and proofs can be found in [6].
Proposition 3. For any Lubin--Tate polynomial f e O^ii^jj and for any nonzero element 9 e OK there exists a unique series [Ojf e O^ii^jj, satisfying two properties:
1. [9jf = 9 ■ X (mod deg 2);
2. f o [% = [d]f o f.
Moreover, for arbitrary non-zero 9l and d2 of OK holds:
• [Q\]f +ff [^2]/ = [Q\ + ^2]/;
• [Oi]f o [$2]f = [di • 62]f ;
• f = M/.
Thus {[a]f, a G OK} is isomorphic to OK as OK-module with the addition in the form of a substitution in the Ff and multiplying by scalar in the form of taking composition with corresponding series.
For a Lubin-Tate polynomial f G OK[X] we denote f o f o • • • o f by fm. Let ^f,m be the set of
m times
roots fm, Fp be the decomposition field of fm over F. Then /j,f,m is a OK-module with addition in the form of a substitution in Ff and multiplication by scalars as a composition with [•]/. Moreover,
Before introducing the Artin map let us put all necessary facts in the next proposition. Proposition 4. • For any a G yf m the map:
a ^ [a]f (a),
is an isomorphism of OK-modules. If a G ^fm, then Fp = F(o) ^FJ/f(—'a) = ^m-l(n).
• a is the uniformizer of Fp and (Ff/F) is a completely ramified Galois extension of degree qm-l(q — 1).
• The following isomorphism is defined:
pftm : Gal(F7/L) = Auto*(pf,m) = (O*/p£)f (a ^ M(a) V a G ^ f,m) ^ u (mod p#).
• Ff is a Galois extension over K. For any a G yf m, the following map is bijective:
Kf /(1+ p£) ^ |J
jez
x (mod (1 + p^)) ^ [xnj]f (a), vK(x) = —j.
• The map pf,m mentioned above continues to isomorphism:
w(Ff/K) ^ Kf/(1 + p£) (a ^ [xnj](a)) ^ x (mod (1 + p^ )), vK(x) = —j. Now if we define F/T = U Fp, then passing to projective limit, we get:
pf : W(FLT/K) = Kf. And we immediately can define the Artin map as the inverse of the map pf :
Artz : Kf ^ W (FLT/K).
Projective line over Fg
Let X be an arbitrary projective smooth irreducible curve over Fg. We denote its field of rational functions by K(X). Let R be a discrete valuation ring containing Fg and whose field of quotients is isomorphic to K(X). Then the unique maximal ideal P C R is called the simple ideal of K(X). The maximal ideal P defines the valuation on the field K (X).
Then we say that:
• f (X) £ K(X) has a pole in P, if uP(f) < 0.
• f (X) £ K(X) has zero in P, if uP(f) > 0.
• The degree of a prime P is the dimension of the field of quotients R/P over Fg.
Let L be a finite extension of the field K (X). We say that the ideal B C L lies above the ideal P C K(X), if Op = K(X) n OB and P = Op n B. The concepts of ramification and inertia are introduced as in the classical case. We assume that an extension over K(X) is unramified if it is unramified in all ideals.
Let us fix some simple ideal of K(X) and denote it by to. Let A := [f (X) £ K(X) | f has no poles in ideals other than to}, then A is the Dedekind integral domain [8]. It is well-known that all the remaining simple ideals of K(X) are in one-to-one correspondence with the simple ideals A[8,p.219].
From now we consider the projective line PFq over Fg. Consider any point of degree one and call such an ideal to. For example, the ideal (1) in the ring Fg [ 1 ]. Then Fg [T] is such a subring of Fg(T), whose elements have poles only at the point to. All other points are in one-to-one correspondence with the unitary irreducible polynomials Fg [T].
Carlitz Module
Our goal is to find a connection between maximal abelian extenstion constructions for local and global geometric fields. As we saw, the Lubin-Tate modules are very useful for local geometric fields. Now we introduce the Carlitz Module the global analogue.
Let L be an arbitrary field. A polynomial h £ L[X] is called additive if
h(X + Y) = h(X)+ h(Y).
A polynomial h £ L[X] is called Fg-linear if it is additive and for any a £ Fg
h(aX) = ah(X).
If the field L is of characteristic p and L contains Fg, then the set of Fg-linear polynomials coincides with the set of polynomials from Xq. On the set of Fg-linear polynomials, can be introduced the ring structure with respect to addition and composition operations. Such a ring is denoted by L(t}, where by t we mean Xq.
For convenience we give an alternative ring construction of L(t}: as a set, it is isomorphic to L[X]; the addition operation is determined by coefficient-wise addition at t in equal degrees; the operation of multiplying two polynomials is a "twisted multiplication": for any a £ L the following is true t ■ a = aq ■ t.
It is easy to see that L(t} is a Fg-algebra. If R is a subring of L, is closed under multiplication by elements of Fq, then R(t} is a Fq-subalgebra of L(t}.
Definition 2. Let L = Fq(T) and R = Fq[T].
Consider the morphism of Fg-algebras:
C : Fq [T] ^ Fq (T)(t}
C : T^ T ■ t 0 + t C : 1 ^ r0 C : 0 ^ 0.
The image of Fg [T] in Fg(T)(t} is called the Carlitz module. The image of an arbitrary element a £ Fg [T] under the action of C is denoted by Ca.
As before we need to look through some statements to use them further. All technical details and proofs can be found in [8, 12 section]. For convenience we collected all necessary statements in one proposition.
We fix a unitary irreducible polynomial P £ Fg [T] of degree dp.
Proposition 5. 1. Cp is the Eisenstein unitary polynomial in Fg[T] of degree qdp with respect to the ideal (P).
2. Cpe is a separable polynomial.
3. Denote the set of zeros of the polynomial Cpe (X) in the algebraic closure1 of Fg(T) by Ape. Then on Ape one can introduce the structure of a Fg [T]-module with standard addition and multiplication by a scalar a £ Fg[T] as an application of Ca.
4. The set Ape is isomorphic, as Fg [T]-module, to Fg \T]/(Pe^Fq [T]). Moreover, for any m £ Fg [T] it is true that the set Am is isomorphic, as Fg[T]-module, to the set Fg[T]/(m ■ Fg[T]), where Am is the set of zeros of the polynomial Cm[X].
5. Denote Fg(T)(Am) by Km. Then the extension Kpe/Fg(T) is a Galois extension with a Galois group:
Gal(Kpe/Fq(T)) = (Fq[T]/(Pe ■ F,[T]))x.
6. If (Q) = (P) and (Q) = to, where to is some simple ideal of Fg(T), (see the previous section) then Kpe is unramified in (Q).
7. Kpe is completely ramified in (P) and, if X is an arbitrary generator of Ape, and Oxpe is the integral closure of Fg [T] in Kpe, then:
(\)(qdP ^^^ ■ OKpe =(P) ■ OKpe .
8. OKpe = Fg [T](A). Main results
Connection between the Lubin—Tate theory and the Carlitz module
In this section we consider projective line PFq. For this section we fix the notation:
• K := Fq (T) is a field of functions of PF q.
• to := () is an ideal in the subring Fg[^] that corresponds to one fixed point on our line.
throughout the paper, the algebraic closure is fixed F q(T).
• А := Fq[Т] is {f (X) е К(X) | f has no poles in ideals other than те}.
• K := Fq((T)) is the completion of the К with respect to ideal (T).
• F := Fq((T))ur is the maximal unramifed extension of K.
Лемма 1. Consider the polynomial f = T ■ X + Xq. Such a polynomial is the Lubin-Tate polynomial for the local field K = Fq((T)). The Lubin-Tate law corresponding to it is the additive law Fadd = X + Y.
Since Fq[T] С OK, it means that for any а е Fq[T] the series [a] f е Of is defined. It is claimed that for any а е Fq [T] :
Hf = С a,
where Ca is considered as an element of Fq(T)[X].
те
Moreover, let а е OK, a = ai ■ Tг, Щ е Fg. Then
i=0
[a] f = ai ' CTi.
i=0
Proof. Note that T is uniformizer for Fq((T)). Also, the residue field of this local field is isomorphic to Fq. From these two remarks it follows that Ct is the Lubin—Tate polynomial for Fq((T)). According to proposition 2, Fadd = F^. Further, it is seen that:
f o Fadd = T • (X + Y) + (X + Y)q,
which is equal to:
Fadd o f = T • X + Xq + T • Y + Yq. The series [a]f is uniquely defined by two conditions:
1. [a]f = aX (moddeg 2);
2. f o [a]f = [a]f o f.
First, check the first condition. It is enough to note that when two polynomials are added, the coefficients at X are added, and during composition are multiplied.
For any a G Fq [T], the polynomial Ca lies in Fq(T)[X], which means Ca = C^. The map C i
q ^^ J, ^wij numiai hcd iu n1q j j , vv niv^n mcdiiD ^a
a morphism of Fq-algebras, and Fq[T] is a commutative ring, which means:
Ct ° Ca = Ca o CT .
Now we notice that f = Ct and get:
f o Ca = Ca o f. It is remaining to prove that if a = ^ ai ■ Tl, ai £ Fq, then
is
i=0
те
[a] f = ^2 ai ' CTi.
i=0
<x
First, check that the series ^ ai • CTi is set correctly. Consider the coefficient bqr of this series
i=0
at XqT. Let us prove that for Tk in bqr there is a finite sum of elements of the field Fq.
Ct" by definition is equal to n (T ■ t0 + t), where the multiplication is twisted. If r < n, then t r = Xq and Xq
is in Ct" with a coefficient divisible by at least Tn r, since when opening the brackets, we must take T ■ t0 no later than in (n — r) — th bracket. Moreover, T ■ t0, and t increase the degree of T by at least 1. Therefore, if n > r + k, then in Cn at XqT there are no terms whose valuation is less or equal to k. That is, for Tk in XqT there is a finite sum of elements from Fç.
Note that E
■ ct
G OK[[X]], which means:
i=0
<x <x
&Î • Cpi = ^ ai • Cpi ) д.
i=0 i=0
Now we are going to check two properties defining the series [a]f.
The first property follows automatically from the fact that CTi = TlX (mod deg 2).
For the second property, we should understand that:
• f ° ^ ai ■ C^) = E ai ■ (f ° ),
i=0 i=0
<x <x
• (E ai ■ &Ti) o f = E ai ■ (CTi O f).
i=0 i=0
Note that both properties are true if we consider expressions modulo deg n, since both series are polynomials of finite degree, and f is Fg—linear. So both equities are true for these series.
Using the fact that f and CTi commute, then equating the right-hand sides of the last two expressions, we obtain the required property. □
We have already noticed that all points of our projective line except infinity are in one-to-one correspondence to unitary irreducible polynomials of Fq[Tj. The local field of the projective line at a point (T) is just Fg(T). The next proposition explores these fields for other points of PF except infinity.
Proposition 6. Let P e Fq[Tj be an arbitrary unitary, irreducible polynomial from T, of degree dp. Then the localization residue field of the ring Fg [Tj by the simple ideal (P) is FgdP.
Proof. Localization by the ideal (P) consists of rational functions of T whose denominators are coprime to P. The only maximum ideal of this ring is P ■ Fg [T](p). By definition, the residue field is Fq[T](P)/(P ■ Fg[T](P)). Let us prove that it is isomorphic to Fq[Tj/(P ■ Fg[Tj).
(P) is the maximum ideal in Fg[Tj, then Fg[T]/(P ■ Fg[Tj) is a field. Consider an arbitrary element ^ e Fg[Tj(P)/(P ■ Fg[Tj(P)), where g(T) is coprime with P. If:
h(T) = qh(T) ■ P + rh(T),
g(T)= qh(T) ■ P + rg(T),
then
w> ^ W>lmoi (P Fim))-
Since Fg[T]/(P ■ Fg[Tj) is a field, then for any rg(T) there exists a polynomial b(T), of degree less than dp, such that g(T) ■ b(T) = 1 (mod (P ■ Fg[Tj)). Thus, Fg[Tj(P)/(P ■ Fg[Tj(P)) is exactly the
set of polynomials, which degree less than dP. Let h^T) and hnT) be two arbitrary elements from Fg[Tj(P)/(P ■ Fg[T](p)). Note that: ' 2
h!(T) + h2(T) = hi(T) ■ g2(T)+ h2(T) ■ gi(T)_ h h
him+¡2M = ^ = hi(T) ■bi(T)+h2 ■b2(T).
Similarly:
hCO ^ = (МТ) ^ 6l(T)) ^ (^2(Т) ^ 62(Т))'
Therefore, the map ^ h(T) • b(T) is an isomorphism. □
Note that when we take the completion of the field К with respect to the valuation associated with the ideal (P), the residue field and the uniformizer do not change with respect to this valuation. The proof of this statement can be found in [9, Claim 1.1.3].
Лемма 2. Cp is the Lubin-Tate polynomial for the completion of К with respect to the valuation associated with the simple ideal (P).
Proof. As already noted in lemma 1, CP = P • X (mod deg 2). Also according to propositions 5, CP is the unitary polynomial of degree qdp, and all its coefficients, except the highest, are divisible by P. We write this in the form of two sequences:
1. Cp = P • X (mod deg 2);
2. Cp = Xqdp (mod (P))'
Note that when the field Fg(T) is the completion with respect to the valuation associated with the ideal (P), the residue field and the uniformizer do not change with respect to this valuation. The proof of this statement can be found in [9, Claim 1.1.3].
Finally, according to the proposition 6 and remark above, the residue field of completion is isomorphic to FgdP, and the maximal ideal is the main ideal (P). □
We fix some irreducible polynomial P £ Fg [Т]. Let fP be the Lubin-Tate polynomial of the field K(P), where K(P) is the completion of К with respect to the valuation, corresponding to the ideal (P)' Let also fP £ O K(p) [X]. Define fP,m as fP о fP о • • • о fP, and K(P),m as the decomposition
m times
field of fP,m over the field K(P). Then the extension tower
K(P) С K(P)y1 С K(p),2 С •• • we call the Lubin-Tate extensions tower with respect to the ideal
oo
(P), and U K(p),m by Lubin-Tate extension of the field К with respect to the ideal (P).
m=1
Теорема 1. The tower of extensions К С Kp С Kp2 С • • • induces the Lubin-Tate extensions tower with respect to the ideal (P). The union of Kpe over all positive integers e induces the Lubin-Tate extension with respect to the ideal (P).
Proof. From the proposition 5, we know that for Kpe there is only one ideal ramified, and it is defined as (A) • OKpe, where OKpe is the integral closure of Fg [T] in KPe.
Consider the ring (OKpe)(д). Since this is the localization of some ring by a simple ideal, (OKpe)(д) is discrete valuation ring with maximum ideal (A). The field of quotients of this ring is the field KPe, since all elements except A and P are already invertible, and the addition of A-1 automatically adds the inverse to P, due to the fact that:
(X)(qdP-1 )-qdP" = (p)'
Therefore, localization by the ideal (A) is the discrete valuation ring that lies above the localization F q [T] by the ideal (P).
We denote the completion of К with respect to the valuation v^p) by K(P) and the completion of KPe with respect to the valuation u^x) by (Kpe)(A). Then, by the proposition 1:
(KPe)(x) = K(P) • kpe = K(P)W,
where A is the primitive root of Cp&.
Since Cp £ Fg[T](t), it means that it lies in p}[X], and therefore for any m :
Cpm = Cp O Cp 0---0 Cp . ' v-*-'
m times
Thus Cpe = Cp,e. It remains to notice that according to the proposition 4,
K(P )(A) = K(P )(Cp, e). Thus, the extension Kpe/k induces on K(P) the Lubin-Tate extension of order e.
oo
Passing to the projective limit with respect to e, we obtain that (J Kpe induces the maximum
e=1
abelian extension of the local field K(P). □
Artin maps for global and local fields
In order to relate theories for the local and global cases, we need first to recall the Artin map in the global case.
Definition 3. Let X be a projective smooth irreducible curve over Fg and K = k(X) be its field of functions. Then:
• For an arbitrary closed point P £ X, we denote the completion of the local ring in P by Op, and the field of quotients of Op by Kp.
• By the idel group Ik of the field K we call bounded product of groups of invertible elements Kp by groups of ring units Op, by all closed points P £ X.
Consider the maximal unramified extension Ku,ab. We call the Artin map [9]:
: Ik/ n ^ Gal(Ku,ab/K) p ex
(... ,ap,...) ^ Frob^dp (ap).
Remark 1. If the extension of the field K is abelian, then for any point (ideal) P we can define the Frobenius automorphism[8, p. 136-137]. It can be described as a mapping Frobp £ Gal(KU,ab/K) such that for any ideal B lying above P it holds:
Frobp(w) = wN(p) (mod B),
where N(P) is the dimension of the residue field of the local ring at the point P over the field Fg.
Consider the local case. Since the maximal unramified abelian extension K, is used, then in the local case it is natural to consider the completion of the maximal unramified extension for Kp, denoting it by Kpr. Then the map Artxp, (remark 9) restricted on Gal(Kpr/K) is defined as:
ArtKp : K* ^ Gal(K]f /K)
X ^ , if Up (x) = j, where 0p is the Frobenius map of the extension K^/K.
Теорема 2. For a partially defined mapping, we introduce the notation aP : Gal(Ku,ab/К) ^ Gal(Kur/К), which sends FrobP to ф—1. Then the following commutative diagram holds:
lp f°p
-Gal(Ku'ab/К)
id
Art
Kp
ap
Gal(K}f /KP)
Proof. ArtKp |cai( Ky/ k) converts any element from Dp to one, since for any x from Dp it is true that up(x) = 0.
Then aP goes to aP(Frob°p^dp(ap)), therefore it goes to $-pordp(ap), which by definition is the image of aP under action of ArtKp lGal(Ky/k). D
Conclusion
In Lemmas 1, 2 and in the theorem 1 it is proved that the extension tower of Carlitz module induces the extension tower of formal Lubin-Tate modules over completion of the local ring at a closed point of the curve.
In the theorem 2 the description was given of the connection between the Artin maps for an arbitrary projective smooth irreducible curve X and the completions of local rings at its closed points. Open questions
• The Carlitz module is a special case of Drinfeld modules of rank 1, which can be considered for an arbitrary smooth projective curve over a finite field. One of the most interesting problems is the generalization of the theorem 1 for an arbitrary smooth projective curve over a finite field.
• As a development of Theorem 1, we can consider a morphism that takes T to an arbitrary polynomial in Fg[T](r), which will be a Lubin-Tate polynomial of degree r. For example, for r = 2, one can send T to the polynomial T ■ X + Xq + T ■ Xq . Then, extending Fg(T) by the powers of this polynomial, we again get a tower of Lubin-Tate extensions for the ideal (T), but it remains unknown what will happen at the other points of our curve.
• Combining the previous two problems leads us to consider Drinfeld modules of rank greater than 1.
СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
1. Востоков С. В. Явная форма закона взаимности //Известия Российской академии наук. Серия математическая. - 1978. - Т. 42. - №. 6. - P. 1288-1321.
2. Востоков С. В. Символ Гильберта в дискретно нормированном поле //Записки научных семинаров ПОМИ. - 1979. - Т. 94. - №. 0. - С. 50-69.
3. Шафаревич И. Р. Общий закон взаимности //Математический сборник. - 1950. - Т. 26. -№. 1. - С. 113-146.
4. Silverman, Joseph H. The arithmetic of elliptic curves. Vol. 106. Springer Science & Business Media, 2009. - Т. 106.
5. Strickland N. P. Formal schemes and formal groups //Contemporary Mathematics. - 1999. - Т. 239. - P. 263-352.
6. Yoshida T. Local class field theory via Lubin-Tate theory //Annales de la Faculte des sciences de Toulouse: Mathematiques. - 2008. - Т. 17. - №. 2. - P. 411-438.
7. Goss D. Basic structures of function field arithmetic. - Springer Science & Business Media, 2012.
8. Rosen M. Number theory in function fields. - Springer Science & Business Media, 2013. - Т. 210.
9. Peter Toth. Geometric abelian class field theory. - Master Thesis, Universiteit Utrecht, 2011.
10. Ивасава К. Локальная теория полей классов. - Мир, 1983.
11. Thakur D. S. Function field arithmetic. - World Scientific, 2004.
REFERENCES
1. Vostokov, S.V., 1978, "Explicit form of the law of reciprocity Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, vol. 42, no. 6, pp.1288-1321.
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4. Silverman, J.H. 2009, The arithmetic of elliptic curves (Vol. 106), Springer Science & Business Media.
5. Strickland, N.P., 1999, "Formal schemes and formal groups Contemporary Mathematics, vol. 239, pp.263-352.
6. Yoshida, T., 2008, "Local class field theory via Lubin-Tate theory In Annales de la Faculté des sciences de Toulouse: Mathématiques, vol. 17, no. 2, pp. 411-438.
7. Goss, D. 2012, Basic structures of function field arithmetic, Springer Science & Business Media.
8. Rosen, M. 2013, Number theory in function fields (Vol. 210), Springer Science & Business Media.
9. Toth, P., 2011, "Geometric abelian class field theory Master Thesis, Universiteit Utrecht.
10. Iwasawa K. and Iwasawa K. 1986, Local class field theory, New York : Oxford University Press.
11. Thakur, D.S. 2004, Function field arithmetic, World Scientific.
