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City University of New York (CUNY) City University of New York (CUNY)

CUNY Academic Works CUNY Academic Works

Dissertations, Theses, and Capstone Projects CUNY Graduate Center

9-2016

Explicit Reciprocity Laws for Higher Local Fields Explicit Reciprocity Laws for Higher Local Fields

Jorge Florez

The Graduate Center, City University of New York

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Explicit Reciprocity Laws for Higher Local Fields

by Jorge Florez

A dissertation submitted to the Graduate Faculty in Mathematics in partial fulfill- ment of the requirements for the degree of Doctor of Philosophy, The City University of New York.

2016

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ii

2016 c

Jorge Florez

All Rights Reserved

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iii This manuscript has been read and accepted for the Graduate Faculty in Mathe- matics in satisfaction of the dissertation requirements for the degree of Doctor of Philosophy.

Victor Kolyvagin

Date Chair of Examining Committee

Ara Basmajian

Date Executive Officer

Victor Kolyvagin

Carlos Moreno

Kenneth Kramer

Supervisory Committee

THE CITY UNIVERSITY OF NEW YORK

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iv Abstract

Explicit Reciprocity Laws for Higher Local Fields

by Jorge Florez

Advisor: Victor Kolyvagin

In this thesis we generalize to higher dimensional local fields the explicit reciprocity laws of Kolyvagin [14] for the Kummer pairing associated to a formal group.

The formulas obtained describe the values of the pairing in terms of multidimensional

p-adic differentiation, the logarithm of the formal group, the generalized trace and

the norm on Milnor K-groups.

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Acknowledgements

I would like to start by thanking my adviser Victor Kolyvagin for suggesting the problem for this thesis and directions for future research. I am also thankful to him for sharing his mathematical ideas, and for the time he spent going over the manuscript providing valuable feedback to improve it. Moreover, I thank the committee members, Kenneth Kramer and Carlos Moreno for taking the time to read my paper and providing feedback.

I want to add an especial acknowledgment to Carlos Moreno for his friendship and for the guidance, support and motivation throughout my professional career.

I would also like to thank many people who were important in the culmination of this project. Among them Professor Jozef Dodziuk and Robert Landsman for their assistance and advise throughout these years. The mathematicians Jorge Basilio, Cihan Karabulut, Tian An Wong and Gabriel Zapata for their warm friendship.

The afternoons we spent talking about mathematics had a great impact in my life;

their constant encouragement made this reached goal possible.

Finally, I want to express my gratitude to my dear family for their love and

v

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vi

support in this important phase of my life.

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For the convenience of the reader we will list some terminology and notation that we will use through out the paper. The letters K, L, M , will be used for one dimensional local fields and the corresponding script capital Latin symbols K, L and M will denote higher local fields containing the corresponding local fields.

• p denotes a fixed prime number.

• Let K/Q

p

denote a local field with ring of integers O

_K

.

• F = F (X, Y ) ∈ O

_K

[[X, Y ]] will denote a formal group of finite height h. Let End

O_K

(F ) be the ring of endomorphisms of F and c : End

O_K

(F ) → O

_K

the embedding g 7→ c(g) = g

⁰

(0).

• Let S be a local field with ring of integers C such that C ⊂ c(End

_O_K

(F )). We will fix a uniformizer π of C. Given a ∈ C, the endomorphism aX + · · · will be denoted by [a]

_F

. For convenience, [π]

_F

will be denoted by f . Thus f

⁽ⁿ⁾

, the nth fold of f , corresponds to [π

ⁿ

]

_F

.

• Let κ

_n

' (C/π

ⁿ

C)

^h

be the π

ⁿ

th torsion group of F and let κ = lim ←− κ

_n

' C

^h

.

Let {e

ⁱ

}

^h_i=1

be a basis for κ and {e

ⁱ_n

}

^h_i=1

be the corresponding reductions to the

group κ

_n

.

(12)

CONTENTS 2

• Let α denote the ramification index of S over Q

p

. We say that a pair (n, t) is admissible if there exist an integer k such that t − 1 − n ≥ αk ≥ n. For example, the pair (n, 2n + α + 1) is admissible with k = [(n + α/α)].

• If R is a discrete valuation ring, the symbols v

_R

, O

_R

, µ

_R

and π

_R

will always denote the valuation, ring of integers, maximal ideal, and some fix uniformizer of R, respectively. Moreover, if the characteristic of the residue field is p, then we define µ

_R,1

= {x ∈ R : v

_R

(x) > v

_R

(p)/(p − 1)}. If R ⊃ K, then F (µ

_R

) and F (µ

_R,1

) will denote the group of µ

_R

-points and µ

_R,1

-points of F , respectively.

• L is a d-dimensional complete field of characteristic 0, i.e., a field for which there are fields L

_d

= L, L

_d−1

, . . . , L

₀

such that L

_i+1

is a complete discrete valuation field with residue field L

_i

, 0 ≤ i ≤ d − 1, and L

₀

is a finite field of characteristic p.

• In what follows we assume that L is a field of mixed characteristic, i.e., char(L) = 0 and char(L

_d−1

) = p. Moreover, we also assume that L ⊃ K(κ

_n

).

• Let t

₁

, . . . , t

_d

be a fixed system of local parameters for L, i.e, t

_d

is a uniformizer for L = L

_d

, t

_d−1

is a unit in O

L

but its residue in L

_d−1

is a uniformizer element of L

_d−1

and so on.

• Let v

L

= (v

₁

, . . . , v

_d

) : L

^∗

→ Z

^d

be the valuation of rank d corresponding

to the system of parameters t

₁

, . . . , t

_d

, i.e., v

_d

= v

L_d

, v

_d−1

(α) = v

L_d−1

(α

_d−1

)

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CONTENTS 3

where α

_d−1

is the residue of α/t

^−v_d ^d^(α)

in L

_d−1

, and so on. Here the group Z

^d

is ordered lexicographically as follows: ¯ a = (a

₁

, . . . , a

_d

) < (b

₁

, . . . , a

_d

) if a

_l

< b

_l

and a

_l+1

= b

_l+1

, ..., a

_d

= b

_d

for some l ≤ d. We denote by Z

^d+

= { (a

₁

, . . . , a

_d

) ∈ Z

^d

: (a

₁

, . . . , a

_d

) > (0, . . . , 0) }.

• O

L

= {x ∈ L : ¯ v

L

(x) ≥ ¯ 0} is the valuation ring of rank d of L, with maximal ideal M

_L

= {x ∈ L : ¯ v

_L

(x) > ¯ 0}. Notice that µ

_L

⊂ M

_L

⊂ O

_L

⊂ O

_L

.

• V

L

= 1 + M

L

is the group of principal units.

• The field L comes equipped with a special topology, the Parshin topology (cf.

§2.4)

• Observe that L

^∗

∼ = V

L

× ht

1

i × · · · × ht

d

i × R

^∗

, where R is a multiplicative closed system of representatives of the last residue field L

₀

and R

^∗

= R{0}.

Then we can endow L

^∗

∼ = V

L

× ht

₁

i × · · · × ht

_d

i × R

^∗

with the product topology, where the group of principal units V

L

has the induced topology from L, and ht

₁

i × · · · × ht

_d

i × R

^∗

with the discrete topology.

• K

_d

(L) is the dth Milnor K-group of the field L, d ≥ 1.

• Let L be a complete discrete valuation field and define

L = L{{T }} = (

_∞

X

−∞

a

_i

T

ⁱ

: a

_i

∈ L, inf v

_L

(a

_i

) > −∞, lim

i→−∞

v

_L

(a

_i

) = +∞

)

.

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CONTENTS 4

Let v

L

(P a

_i

T

ⁱ

) = min

_i∈Z

v

_L

(a

_i

), so O

L

= O

_L

{{T }} and µ

L

= µ

_L

{{T }}. Ob- serve that O

L

/µ

L

= k

_L

((T )), where k

_L

is the residue field of L.

• We define

L{{T

1

}} · · · {{T

d−1

}},

inductively by E

_d−1

{{T }} where E

_d−1

= L{{T

₁

}} · · · {{T

_d−2

}}.

• D(L/K) will denote the different of the the extension of local fields L/K.

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Chapter 1 Introduction

1.1 Background

The history for finding explicit formulations of local class field theory dates back to Kummer [15] in 1858 when he tried to describe the Hilbert symbol, associated to the cyclotomic field Q

p

(ζ

_p

), in terms of logarithmic derivatives. Seventy years later, when class field theory was acquiring its final shape, Artin and Hasse [1] provided an explicit formula for the Hilbert symbol associated to Q

p

(ζ

_p^m

); p being an odd prime.

The formulas of Artin-Hasse would find an application in the work of Iwasawa [10] when he studied the units of the cyclotomic field Q(ζ

^p^m

). Iwasawa extended the formulas of Artin-Hasse to describe more values of the Hilbert symbol (cf. [9]), in terms of p-adic logarithmic derivatives. This would be the first manifestation of a deep connection, via explicit reciprocity laws, between arithmetical objects and values of zeta functions. The work of Iwasawa inspired Coates and Wiles [3] to do fundamental work on the Birch and Swinnerton Dyer conjecture. One of the main

5

(16)

CHAPTER 1. 6

tools in their paper [27] was to extend Iwasawa’s reciprocity laws to Kummer pairings associated to Lubin-Tate formal groups, which are analogues of the Hilbert symbol for formal groups.

Soon after, Kolyvagin [14] extended the formulas of Wiles from Lubin-Tate formal groups to arbitrary formal groups (of finite height) and described the Kummer pairing in terms of p-adic derivations. Moreover, Kolyvagin also showed that the pairing is completely characterized by its values at the torsion points of the formal group.

Parallel to this, the theory of higher local fields was developing and along with it was the class field theory for such fields. Different formulations for a such theory were proposed by many people but it would be Kato’s [11] (see also [22]), who by means of Milnor K-groups, developed the most commonly used theory. Kato’s formulation shows that the finite abelian extensions of a d-dimensional higher local field correspond to norm subgroups of its Milnor dth K-group. This correspondence was established via a reciprocity map between the dth Milnor K-group and the galois group of the maximal abelian extension of the higher local field. I. Fesenko [4] nicely contributed to the class field theory for higher local fields also.

After Kato developed this class field theory, the program to make this theory

more explicit began. Some of the higher dimensional explicit reciprocity laws that

can be found in the literature are those of Kurihara [16], Vostokov [25], [26], Zinoviev

[28] and Kato [12]. These explicit laws describe the analogue of Hilbert symbol for

(17)

CHAPTER 1. 7

higher local fields, in terms of higher dimensional p-adic logarithmic differentiation.

Using his explicit reciprocity laws, Kato made fundamental progress on Iwasawa main conjecture for modular forms. Kato’s explicit laws provide a link between the Euler system of Beilinson elements and some special values of the L-function attached to a modular form. This allowed Kato [13] to prove a divisibility statement related to the Iwasawa main conjecture for modular forms.

In this thesis, we will derive explicit reciprocity laws for the higher dimensional analogue of the Kummer pairing associated to a formal group F , or generalized Kummer pairing. The construction of these laws involve higher dimensional p-adic derivations, the logarithm of the formal group F , the generalized trace and the norm of Milnor K-groups. These formulas constitute a generalization, to higher local fields, of the explicit reciprocity laws of Kolyvagin [14]. In particular, the formulas apply to the generalized Hilbert symbol, like those of Kurihara [16] and Zinoviev [28]. It is still a work in progress to determine if the formulas of these two authors can be derived from the explicit reciprocity laws constructed in this thesis.

Finally, it is worth mentioning that the techniques that are used to obtain the

explicit reciprocity laws are inspired by the work of Kolyvagin in [14]. In particular,

this allows for a more classical and conceptual approach to the higher dimensional

reciprocity laws.

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CHAPTER 1. 8

1.2 Description of the problem and results

Let K/Q

^p

be a local field and F a one-dimensional formal group of finite height with coefficients in the ring of integers of K. Let End

_O_K

(F ) be the ring of endomorphisms of F and c : End

_O_K

(F ) → O

_K

the embedding f 7→ c(f ) = f

⁰

(0). Let S/Q

p

be a local field with ring of integers C such that C ⊂ c(End

O_K

(F )) ⊂ O

_K

. Let us fix a uniformizer π for S and denote by f = [π]

_F

the endomorpshism such that c(f ) = π.

Let h be the height of the endomorphism [π]

_F

with respect to C, i.e, [π]

_F

(X) = g(X

^q^h

) (mod πC) where g

⁰

(0) 6= 0 (mod πC) and q = |k

_S

|. Denote by f

⁽ⁿ⁾

= [π

ⁿ

]

_F

the n-fold of f . Let κ

n

be the π

ⁿ

th torsion group, i.e., the set of those v ∈ K such that f

⁽ⁿ⁾

(v) = 0. Let {e

ⁱ_n

}

^h_i=1

be a C/π

ⁿ

C-basis for the group κ

_n

' (C/π

ⁿ

C)

^h

.

Let L ⊃ K be a local field containing κ

_n

. Let O

_L

be the ring of integers of L, µ

_L

its maximal ideal, v

_L

its valuation and π

_L

be a uniformizer for L. Define

L = L{{T

₁

}} · · · {{T

_d−1

}},

inductively. The ring on integers and maximal ideal are, respectively,

O

_L

= O

_L

{{T

₁

}} · · · {{T

_d−1

}} and µ

_L

= µ

_L

{{T

₁

}} · · · {{T

_d−1

}}.

We define the canonical pairing (cf. § 4.1)

(, )

L,n

: K

_d

(L) × F (µ

L

) → κ

_n

,

(19)

CHAPTER 1. 9

by

(α, x) 7→ Υ

L

(α)(z) −

_F

z,

where K

_d

(L) is the dth Milnor K-group of L, Υ

_L

: K

_d

(L) → G

^ab_L

is the Kato’s reciprocity map for L, f

⁽ⁿ⁾

(z) = x and −

_F

is the subtraction in the formal group F , cf. §4.1. Denote by (, )

ⁱ_L,n

the ith coordinate of (, )

L,n

with respect to the basis {e

ⁱ_n

}.

In this paper we will use the technique of Kolyvagin in [14] to derive explicit formulas for the symbol (, )

ⁱ_L,n

in terms of the generalized trace, the norm of Milnor K-groups, d-dimensional derivations and the logarithm of the formal group.

More specifically, let E be a complete discrete valuation field. We define a map

c

_{E{{T }}/E}

: E{{T }} → E

by c

_{E{{T }}/E}

( P

i∈Z

a

_i

T

ⁱ

) = a

₀

. Then we can define c

L/L

by the composition c

_L{{T₁}}/L

◦ · · · ◦ c

L/L{{T₁}}...{{T_d−2}

We define the generalized trace by T

^L/S

= Tr

_L/S

◦ c

L/L

.

The main result in this paper is the following (cf. Theorem 7.3.1). Let M = L(κ

_t

), t ≥ m ≥ n, for t large enough with respect to m and put M = M {{T

₁

}} · · · {{T

_d−1

}}, then

(N

M/L

{a

₁

, . . . , a

_d

}, x)

ⁱ_L,n

= T

^M/S

D

ⁱ_M,m

(a

₁

, . . . , a

_d

) a

₁

· · · a

_d

l

_F

(x)

for all a

1

, . . . , a

d

∈ M

^∗

(see (7.1) where cases when a

i

are units or not are considered)

(20)

CHAPTER 1. 10

and all x ∈ F (µ

L

). Here l

_F

is the formal logarithm, N

M/L

is the norm on Milnor K-groups and D

_M,mⁱ

is a d-dimensional derivation of O

L

over O

_K

, i.e.,

D

_M,mⁱ

: O

^d_M

→ W

is a map, with W an O

_M

-module, satisfying the properties in definition 6.2.1, that is, for all a

₁

, . . . , a

_d

and a

⁰₁

, . . . , a

⁰_d

in O

M

we have:

1. Leibniz rule:

D(a

₁

, . . . , a

_i

a

⁰_i

, . . . , a

_d

) = a

⁰_i

D(a

₁

, . . . , a

_i

, . . . , a

_d

) + a

_i

D(a

₁

, . . . , a

⁰_i

, . . . , a

_d

),

for any 1 ≤ i ≤ d.

2. Linearity:

D(a

₁

, . . . , a

_i

+ a

⁰_i

, . . . , a

_d

) = D(a

₁

, . . . , a

_i

, . . . , a

_d

) + D(a

₁

, . . . , a

⁰_i

, . . . , a

_d

),

for any 1 ≤ i ≤ d.

3. Alternate: D(a

₁

, . . . , a

_i

, . . . , a

_j

, . . . , a

_d

) = 0, if a

_i

= a

_j

for i 6= j.

4. D(a

1

, . . . , a

d

) = 0 if some a

i

∈ O

K

.

Notice that D is a 1-dimensional derivation in each component and moreover, it

can be parametrized by 1-dimensional derivations. Indeed, just as one dimensional

derivations are parametrized by Ω

O_K

(O

L

), the O

L

-module of Khaler differentials of

(21)

CHAPTER 1. 11

O

L

over O

_K

, then it will also be shown in §6.2 Proposition 6.2.1 that all such d- dimensional derivations are parametrized by the O

L

-module V

d

Ω

O_K

(O

L

), i.e., the d-th exterior product of Ω

O_K

(O

L

). This is the O

L

-module Ω

O_K

(O

L

) ⊗ · · · ⊗ Ω

O_K

(O

L

) divided out by the O

L

-submodule generated by the elements

x

₁

⊗ · · · ⊗ x

_d

where x

_i

= x

_j

for some i 6= j, where x

₁

, . . . , x

_d

∈ Ω

O_K

(O

L

).

Moreover, notice that from properties (2), (3) and (4) it easily follows that D satisfies the relations

D(a

1

, . . . , a

i

, . . . , a

j

. . . , a

d

) = 0, (1.1)

if a

_i

+ a

_j

= 1, for i 6= j, which are called the Steinberg relations. Therefore, following Kolyvagin [14], we define the map

ψ : (O

^×_M

)

^d

→ W

by

{a

₁

, . . . , a

_d

} → D

ⁱ_M,m

(a

1

, . . . , a

d

) a

₁

. . . a

_d

,

This map can be extended to all of K

_d

(M

^∗

) by equation (7.1) from Section 7.1.

As in the work of Kolyvagin [14], these derivations will be normalized in terms of the invariants (7.7) attached to the Galois representation on the Tate-module (7.5).

More specifically, we will build the d-dimensional derivation D

ⁱ_M,m

out of the

(22)

CHAPTER 1. 12

condition

D

ⁱ_M,m

(T

₁

, . . . , T

_d−1

, e

^j_t

) = − T

₁

· · · T

_d−1

c

_i,j

l

⁰

(e

^j_t

) , cf. Proposition 7.2.3, where c

_i,j

is defined in equation (7.7).

The deduction of the formula starts by proving that the pairing (, )

L,n

is sequentially continuous in the second argument with respect to the Parshin topology, i.e., if α ∈ K

_d

(L) and {x

_k

} is a sequence in F (µ

L

) such that x

_k

→ 0 then (α, x

_k

)

L,n

→ 0, cf. Proposition 4.2.2, which is vital for the rest of the proof and was inspired by the proof of Proposition 4.1 in [14]. From this, we can prove the existence of the so called Iwasawa maps, ψ

_L,nⁱ

: K

d

(L) → R

L,1

/π

ⁿ

R

L,1

(cf. Proposition 5.2.1), such that

(α, x)

ⁱ_L,n

= T

L/S

( ψ

_L,nⁱ

(α) l

_F

(x) ) ∀x ∈ F (µ

L,1

),

where F (µ

L,1

) is the set {x ∈ L : v

L

(x) ≥ [v

L

(p)/(p − 1)] + 1} considered with the operation defined by F . We then define the maps D

ⁱ_L,n

out of ψ

ⁱ_L,n

as in equation (5.17). The main goal is to prove that under certain conditions the map D

ⁱ_M,m

is a d-dimensional derivation over O

K

, cf. Proposition 7.1.1. A vital role in this paper is given by the norm series relations

({r(x), a

₂

, . . . , a

_d

}, x)

L,n

= 0, ∀x ∈ F (µ

L

), ∀a

₂

, . . . , a

_d

∈ L

^∗

,

cf. Proposition 4.3.1. These series allow us to obtain the representation (5.8) in

Proposition 5.2.3 which together with the continuity of the pairing we obtain the

(23)

CHAPTER 1. 13

identity (5.18) from Proposition 5.4.2. But Proposition 5.4.4 tells us that every element in µ

L

has a representation of the form η(T

₁

, . . . , T

_d−1

, π

_L

), where η(X

₁

, . . . , X

_d

) is a power series in the variables X

₁

, . . . , X

_d

with coefficients in O

_L

, given by equation (5.4.5).

Then, we will conclude from Proposition 5.4.5 and corollary 5.4.4 that

D

_L,mⁱ

(α

₁

, . . . , α

_d

) = det ∂η

_i

∂X

_j

i,j

Xk=Tk,

k=1,...,d

D

_L,nⁱ

(T

₁

, . . . , π

_L

),

where α

_i

= η

_i

(T

₁

, . . . , π

_L

), i = 1, . . . , d and T

_d

= π

_L

. This last equation tells us that D

_L,nⁱ

behaves like a d-dimensional derivation when restricted to µ

L

.

This will be used in the proof of Proposition 7.1.1 to conclude that D

_M,mⁱ

, for M = M {{T

1

}} · · · {{T

_d−1

}} and M a finite extension of L containing κ

_t

for large enough t with respect to m, is a d-dimensional derivation when restricted to certain quotient of M as an O

M

-module, namely R

M,1

/(π

^m

/π

_M

)R

M,1

. Moreover, Proposition 7.2.3 gives us a way of constructing explicitly this derivation out of the condition

D

ⁱ_M,m

(T

₁

, . . . , T

_d−1

, e

^j_t

) = −T

₁

· · · T

_d−1

c

_i,j

/l

⁰

(e

^j_t

).

We can express back the Iwasawa map ψ

_M,mⁱ

: K

d

(M) → R

M,m

/π

^m

R

M,m

in terms of the derivation D

ⁱ_M,m

as

ψ

_M,mⁱ

({a

₁

, . . . , a

_d

}) = D

ⁱ_M,m

(a

₁

, . . . , a

_d

)

a

1

· · · a

_d

(24)

CHAPTER 1. 14

cf. 7.1. Finally, the return to the field L will be guaranteed by Proposition 7.1.2.

The final formulas are contained in Theorem 7.3.1.

In chapter 8 we specialize to the case of a Lubin-tate formal group and refine Theorem 7.3.1 for this case. In particular, we give an explicit computation of the invariants c

_i,j

, namely

c

_i,j

= − 1

π

^t

.

(25)

Chapter 2 Higher-dimensional local fields

2.1 2-dimensional local fields

We will start defining 2-dimensional local fields since most of the proofs will be reduced to this case. In the next sections all the concepts will be generalized to higher dimensions.

Definition 2.1.1. We say that K is a 2-dimensional local field if there are fields K

2

= K, K

1

, and K

0

such that K

i+1

is a complete discrete valuation ring with residue field K

_i

, 0 ≤ i ≤ 1, and K

₀

is a finite field of characteristic p.

We have the following examples of 2-dimensional local fields:

1. F

^q

((T

1

))((T

2

)), where q is a power of the prime p. In this case K

2

= F

^q

((T

1

))((T

2

)), K

₁

= F

q

((T

₁

)) and K

₀

= F

q

. Recall that if k is a field then k((T )) is the fraction field of the ring k[[T ]].

2. E((T )), where E is a finite extension of Q

p

. Here K

₁

= E and K

₀

= k

_E

15

(26)

CHAPTER 2. HIGHER-DIMENSIONAL LOCAL FIELDS 16 3. For a complete discrete valuation field E we define

K = E{{T }} = (

_∞

X

−∞

a

_i

T

ⁱ

: a

_i

∈ E, inf v

_E

(a

_i

) > −∞, lim

i→−∞

v

_E

(a

_i

) = +∞

)

Let v

K

(P a

i

T

ⁱ

) = min v

E

(a

i

). Then K is a complete discrete valuation field with residue field k

_F

((t)) and ring of integers

O

_{E{{T }}}

= (

_∞

X

−∞

a

_i

T

ⁱ

∈ F {{T }} : a

_i

∈ O

_F

, ∀i ∈ Z )

Thus, if E is a local field then E{{T }} is a 2-dimensional local field with K

₁

= k

_E

((t)) and K

₀

= k

_E

. These fields are called standard 2-dimensional local fields (cf. [7] §1.1).

We are interested only in the case where K is of mixed characteristic, i.e., char(K

₁

) = p. The following Theorem classifies such fields.

Theorem 2.1.1 (Classification Theorem). Let K be a 2-dimensional local field of mixed characteristic. Then K is a finite extension of a 2-dimensional standard field E{{T

₁

}}, where E is a finite extension of Q

p

, and there is a finite extension of K which is a standard 2-dimensional local field.

Proof. cf. [7] § 1.1 Classification Theorem.

We define the ring of integers of K to be the set

O

K

= { x ∈ O

K

: ¯ x ∈ O

_K₁

}

(27)

CHAPTER 2. HIGHER-DIMENSIONAL LOCAL FIELDS 17

and the maximal ideal

M

K

= { x ∈ O

K

: ¯ x ∈ µ

_K₁

} where ¯ x denotes the reduction of x in K

₁

. Notice that

µ

K

⊂ M

K

⊂ O

K

⊂ O

K

.

In the examples above we can compute these two sets explicitly:

1. If K = F

q

((T

₁

))((T

₂

)): O

K

= F

q

[[T

₁

]] + T

₂

F

q

((T

₁

))[[T

₂

]] and M

K

= T

₁

F

q

[[T

₁

]] + T

₂

F

q

((T

₁

))[[T

₂

]]

2. If K = E((T )): O

K

= O

_E

+ T E[[T ]] and M

K

= π

_E

O

_E

+ T E[[T ]].

3. If K = E{{T }}: O

K

= P

∞

−∞

a

_i

T

ⁱ

∈ K : a

_i

∈ O

_E

, ∀i ≥ 0 and a

_i

∈ µ

_E

∀i < 0 and M

K

= P

∞

−∞

a

i

T

ⁱ

∈ F {{T }} : a

i

∈ O

E

, ∀i > 0 and a

i

∈ µ

E

∀i ≤ 0

In particular, if K = Q

p

{{T }} we have

O

_K

= {

∞

X

−∞

a

_i

T

ⁱ

∈ K : a

_i

∈ Z

p

, ∀i ≥ 0 and a

_i

∈ pZ

p

∀i < 0},

and

M

K

= {

∞

X

−∞

a

_i

T

ⁱ

∈ K : a

_i

∈ Z

p

, ∀i > 0 and a

_i

∈ pZ

p

∀i ≤ 0}.

Topology on K: The topology on K = E{{T }} is called the Parshin topology

(28)

CHAPTER 2. HIGHER-DIMENSIONAL LOCAL FIELDS 18

or Higher-dimensional topology (cf. [7] §1.3 ). A basis of neighborhoods of 0 for this topology is given by the following: Let {U

_i

}

_i∈Z

be a sequence of neighborhoods of 0 in the local field E, i.e., U

_i

= π

_Eⁿⁱ

O

_E

, where n

_i

∈ Z, such that

1. There exist a c ∈ Z such that π

E^c

O

_E

⊂ U

_i

for all i ∈ Z, i.e., ∩U

i

⊃ π

^c_E

O

_E

. This says that the neighborhoods U

_i

’s cannot be arbitrarily small.

2. For every l ∈ Z we have π

E^l

O

E

⊂ U

i

for sufficiently large i. This implies that the U

_i

’s get bigger as i → ∞.

We put

U

{Ui}

= {

∞

X

−∞

a

i

T

ⁱ

∈ K : a

i

∈ U

i

} These sets U

{U_i}

form a base of neighborhoods of 0 in K.

Remark 2.1.1. Observe that in this topology an element x = P

∞

−∞

a

_i

T

ⁱ

∈ K = E{{T }} converges as an infinite sum since both tails P

i>N

a

_i

T

ⁱ

and P

i<−N

a

_i

T

ⁱ

approach to 0 as N → ∞. To see this take a basis element for zero, U

{Ui}

, and let c be given by condition 1 above, and for l = v

_K

(x) = inf v

_E

(a

_i

) let N be large enough to satisfy condition 2 above.

Since by definition v

_E

(a

_i

) → ∞ as i → −∞ we can find an N such that v

_E

(a

_i

) > c for i < −N . Then a

_i

∈ π

^c_E

O

_E

⊂ U

_i

for i < −N , i.e., P

i<−N

a

_i

T

ⁱ

∈ U

{U_i}

.

(29)

CHAPTER 2. HIGHER-DIMENSIONAL LOCAL FIELDS 19 Now v

_E

(a

_i

) ≥ l for all i, then a

_i

∈ π

_E^l

O

_E

⊂ U

_i

for i > N . Again, this implies P

i>N

a

_i

T

ⁱ

∈ U

{U_i}

Remark 2.1.2. Observe also that a sequence x

_n

= P

∞

−∞

a

⁽ⁿ⁾_i

T

ⁱ

converging to zero in this topology does not necessarily satisfy that v

K

(x

_n

) → ∞, i.e., the coefficients a

⁽ⁿ⁾_i

do not necessarily converge uniformly on i to zero. For instance, the sequence x

_n

= T

ⁿ

converges to zero in this topology and v

_K

(x

_n

) = 0 for all n. But the following Proposition is true:

Proposition 2.1.1. The sequence x

n

= P

∞

−∞

a

⁽ⁿ⁾_i

T

ⁱ

converges to zero if and only if the following two conditions are satisfied

(i) inf v

K

(x

_n

) > −∞;

(ii) for a given m ∈ Z

+

the sequence a

⁽ⁿ⁾_i ^n→∞

→ 0 uniformly for every i ≤ m.

That is, the coefficients are uniformly bounded and converge to zero uniformly for i ≤ m, for every m.

Proof. →) Suppose the sequence x

_n

converges to zero. For a fix m ∈ Z and given M > 0 define

U

_i^M

=

( π

_E^M

O

_E

, i ≤ m π

_E⁻ⁱ

O

_E

, i > m Then V

_M

= V

_U^M

i

is a basis element for zero, so there exists an N such that x

_n

∈ V

_M

for n > N . That is v

_E

(a

⁽ⁿ⁾_i

) ≥ M for n > N and all i ≤ m, then a

⁽ⁿ⁾_i ^n→∞

→ 0 uniformly

on every i ≤ m.

(30)

CHAPTER 2. HIGHER-DIMENSIONAL LOCAL FIELDS 20

Let us show now that inf v

_K

(x

_n

) > −∞. For i > 1 define m

_i

= inf

_n

v

_E

(a

⁽ⁿ⁾_i

) and M

_i

= min

_1≤k≤i

{m

_k

}. Then M

_i

≥ M

_i+1

and moreover, if inf v

_K

(x

_n

) = −∞, then M

_i ^i→∞

→ −∞ which follows from the just proven condition (ii). Consider

U

_i

=

( π

_E

O

_E

, i ≤ 0 π

_E^Mⁱ⁺¹

O

_E

, i > 0

Then U

_{U_i_}

is a base element for 0. Again, since we are assuming inf v

_K

(x

_n

) = −∞, for certain large enough i we have the strict inequality M

_i−1

> M

_i

, i.e., M

_i

= m

_i

and so we can find a subsequence x

_n_i

= P

j

a

⁽ⁿ_j ⁱ⁾

T

^j

such that v

_K

(a

⁽ⁿ_i ⁱ⁾

) = M

_i

, thus v

_K

(x

_n_i

) ≤ v

_E

(a

⁽ⁿ_i ⁱ⁾

) = M

_i

. Then a

⁽ⁿ_i ⁱ⁾

6∈ U

_i

, which implies x

_n_i

6∈ U

{U_i}

. Finally, since n

_i

→ ∞ as i → ∞ this contradicts the convergence of x

_n

to zero.

←) Let x

n

be a sequence in K satisfying (i) and (ii). Let U

{Ui}

is a base element for 0. Then there exist a c ∈ Z such that π

E^c

O

_E

⊂ U

_i

for all i ∈ Z. Also for l = inf v

_K

x

_n

there exist an i

₀

such that π

^l_E

O

_E

⊂ U

_i

for all i > i

₀

.

By condition (ii) applied to m = i

₀

there exist an N such that v

_E

(a

⁽ⁿ⁾_i

) > c for all n ≥ N and all i ≤ i

₀

. Then a

⁽ⁿ⁾_i

∈ π

_E^c

O

_E

⊂ U

_i

for all n ≥ N and all i ≤ i

₀

. On the other hand, v

_E

(a

⁽ⁿ⁾_i

) ≥ inf v

_K

(x

_n

) = l, so a

⁽ⁿ⁾_i

∈ π

_E^l

O

_E

⊂ U

_i

for all i > i

₀

. Summarizing we have that a

⁽ⁿ⁾_i

∈ U

_i

for all n > N and all i ∈ Z. Therefore x

n

∈ U

{U_i}

and so it converges to zero in the Parshin topology.

(31)

CHAPTER 2. HIGHER-DIMENSIONAL LOCAL FIELDS 21

Topology on K

^∗

:

Let R ⊂ K = K

₂

be a set of representatives of the last residue field K

₀

. Let t

₁

and t

₂

be a fixed system of local parameters for K, i.e, t

₂

is a uniformizer for K and t

1

is a unit in O

K

but its residue in K

1

is a uniformizer element of K

1

. Then

K

^∗

= V

_K

× ht

₁

i × ht

₂

i × R

^∗

,

where the group of principal units V

K

= 1 + M

K

and R

^∗

= R{0}. From this observation we have the following,

Proposition 2.1.2. We can endow K

^∗

with the product of the induced topology from K on the group V

K

and the discrete topology on ht

₁

i × ht

₂

i × R

^∗

. Moreover, every Cauchy sequence with respect to this topology converges in K

^∗

.

Proof. cf [7] Chapter 1: Higher dimensional local fields.

2.2 d-dimensional local fields

We will now generalize all the concepts introduce before to any dimension. In the rest of this section E will denote a local field, k

_E

its residue field and π

_E

a uniformizer for E.

Definition 2.2.1. K is an d-dimensional local field, i.e., a field for which there is a

chain of fields K

_d

= K, K

_d−1

, . . . , K

₀

such that K

_i+1

is a complete discrete valuation

ring with residue field K

_i

, 0 ≤ i ≤ d − 1, and K

₀

is a finite field of characteristic p.

(32)

CHAPTER 2. HIGHER-DIMENSIONAL LOCAL FIELDS 22 If k is a finite field then K = k((T

₁

)) . . . ((T

_d

)) is a d-dimensional local field with

K

_i

= k((T

₁

)) . . . ((T

_i

)) 1 ≤ i ≤ d.

If E is a local field, then K = E{{T

1

}} . . . {{T

d−1

}} is defined inductively as

E

_d−1

{{T

_d−1

}},

where E

_d−1

= E{{T

₁

}} . . . {{T

_d−2

}}. We have that K is a d-dimensional local field with residue field K

_d−1

= k

_E_d−1

((T

_d−1

)), and by induction

k

_E_d−1

= k

_E

((T

₁

)) . . . ((T

_d−2

)).

Therefore K

d−1

= k

E

((T

1

)) . . . ((T

d−1

)). These fields are called the standard fields.

From now on we will assume K has mixed characteristic, i.e., char(K)=0 and char(K

_d−1

)= p. The following theorem classifies all such fields.

Theorem 2.2.1 (Classification Theorem). Let K be an d-dimensional local field of mixed characteristic. Then K is a finite extension of a standard field

E{{T

₁

}} . . . {{T

_d−1

}},

where E is a local field, and there is a finite extension of K which is a standard field.

Proof. cf. [7] § 1.1 Classification Theorem.

Definition 2.2.2. An d-tuple of elements t

₁

, . . . , t

_d

∈ K is called a system of local

(33)

CHAPTER 2. HIGHER-DIMENSIONAL LOCAL FIELDS 23

parameters of K, if t

_d

is a prime in K

_d

, t

_d−1

is a unit in O

K

but its residue in K

_d−1

is a prime element of K

_d−1

, and so on.

For the standard field E{{T

₁

}} . . . {{T

_d−1

}} we can take as a system of local parameters t

d

= π

E

, t

d−1

= T

d−1

, . . . , t

1

= T

1

.

Definition 2.2.3. We define a discrete valuation of rank d to be the map v = (v

₁

, . . . , v

_d

) : K

^∗

→ Z

^d

, v

_d

= v

K_d

, v

_d−1

(x) = v

K_d−1

(x

_d−1

) where x

_d−1

is the residue in K

_d−1

of xt

^−v_d ⁿ^(x)

, and so on.

Although the valuation depends, for n > 1, on the choice of t

₂

, . . . , t

_d

, it is independent in the class of equivalent valuations.

2.3 Extensions of K

Let L/K be a finite extension of the d-dimensional local field K. Then L is also a d-dimensional local field.

Definition 2.3.1. Let t

₁

, . . . , t

_d

and t

⁰₁

, . . . , t

⁰_d

be a system of local parameters for K and L, respectively, with associated valuations v and v

⁰

, respectively. Put

E(L/K) = (v

_j⁰

(t

_i

))

_i,j

=







e

₁

0 . . . 0 . . . e

₂

. . . 0 . . . . . . . . . 0 . . . . . . . . . e

_d







where e

_i

= e(L

_i

/K

_i

), i = 1, . . . , n. Then e

_i

does not depend on the choice of the

(34)

CHAPTER 2. HIGHER-DIMENSIONAL LOCAL FIELDS 24 parameters, and [L/K] = f (L/K) Q

n

i=1

e

_i

(L/K), where f (L/K) = [L

₀

/K

₀

] (cf. [7]

§1.2).

2.4 Topology on K

We define the topology on E{{T

₁

}} . . . {{T

_d−1

}} by induction on d. For d = 1 we define the topology to be the topology of a one-dimensional local field. Suppose we have defined the topology on a standard d-dimensional local field E

_d

and let K = E

_d

{{T }}. Denote by P

_E_d

(c) the set {x ∈ E

_d

: v

_E_d

(x) ≥ c}. Let {V

_i

}

_i∈Z

be a sequence of neighborhoods of zero in E

_d

such that







1. there is a c ∈ Z such that P

Ed

(c) ⊂ V

_i

for all i ∈ Z.

2. for every l ∈ Z we have P

Ed

(l) ⊂ V

_i

for all sufficiently large i.

(2.1)

and put V

{Vi}

= {P b

i

T

ⁱ

: b

i

∈ V

i

}. These sets form a basis of neighborhoods of 0 for a topology on K. For an arbitrary d-dimensional local field L of mixed characteristic we can find, by the Classification Theorem, a standard field that is a finite extension of L and we can give L the topology induced by the standard field.

Proposition 2.4.1. Let L be a d-dimensional local field of mixed characteristic with the topology defined above.

1. L is complete with this topology. Addition is a continuous operation and mul-

tiplication by a fixed a ∈ L is a continuous map.

(35)

CHAPTER 2. HIGHER-DIMENSIONAL LOCAL FIELDS 25

2. Multiplication is a sequentially continuous map, i.e., if x ∈ L and y

_k

→ y in L then xy

_k

→ xy.

3. This topology is independent of the choice of the standard field above L.

4. If K is a standard field and L/K is finite, then the topology above coincides with the natural vector space topology as a vector space over K.

5. The reduction map O

L

→ k

L

= L

_d−1

is continuous and open (where O

L

is given the subspace topology from L, and k

L

= L

_d−1

the (d − 1)-dimensional topology).

Proof. All the proofs can be found in [20] Theorem 4.10.

2.5 Topology on K ^∗

Let R ⊂ K = K

_d

be a set of representatives of the last residue field K

₀

. Let t

₁

, . . . , t

_d

be a fixed system of local parameters for K, i.e., t

_d

is a uniformizer for K, t

_d−1

is a unit in O

K

but its residue in K

_d−1

is a uniformizer element of K

_d−1

, and so on. Then

K

^∗

= V

K

× ht

₁

i × · · · × ht

_d

i × R

^∗

,

where the group of principal units V

K

= 1 + M

K

and R

^∗

= R{0}. From this

observation we have the following,

(36)

CHAPTER 2. HIGHER-DIMENSIONAL LOCAL FIELDS 26 Proposition 2.5.1. We can endow K

^∗

with the product of the induced topology from K on the group V

K

and the discrete topology on ht

₁

i × · · · × ht

_d

i × R

^∗

. In this topology we have,

1. Multiplication is sequentially continuous, i.e., if a

n

→ a and b

n

→ b then a

_n

b

_n

→ ab.

2. Every Cauchy sequence with respect to this topology converges in K

^∗

.

Proof. cf [7] Chapter 1 §1.4.2.

(37)

Chapter 3 Formal groups

In the rest of this paper F will be a formal group over the ring of integers, O

_K

, of a local field K/Q

p

. Let End

O_K

(F ) be the ring of endomorphisms of F . For t ∈ End

O_K

(F ) we denote t

⁰

(0) by c(t). This induces an embedding

c : End

_O_K

(F ) → O

_K

.

We have that that c(End

O_K

(F )) is a closed subring of O

_K

( cf. [14] §2.3). In particular Z

p

⊂ c(End

O_K

(F )). Let S/Q

p

be a local field with ring of integers C such that C ⊂ c(End

O_K

(F )). Let us fix once and for all a uniformizer π for C. Let j be the degree of inertia of S/Q

p

, i.e., |k

_S

| = p

^j

. In what follows f will denote the element in End

OK

(F ) such that c(f ) = π.

27

(38)

CHAPTER 3. FORMAL GROUPS 28

3.1 The Weiertrass lemma

Let E be a discrete valuation field of zero characteristic with integer ring O

E

and maximal ideal µ

_E

.

Lemma 3.1.1 (Weierstrass lemma). Let g = a

₀

+ a

₁

X + · · · ∈ O

_E

[[X]] be such that a

₀

, . . . , a

_n−1

∈ µ

_E

, n ≥ 1, and a

_n

6∈ µ

_E

. Then there exist a unique monic polynomial c

₀

+ · · · + X

ⁿ

with coefficients in µ

_E

and a series b

₀

+ b

₁

X · · · with coefficients in O

_E

and b

₀

a unit, i.e., b

₀

6= µ

_E

, such that

g = (c

₀

+ · · · + X

ⁿ

)(b

₀

+ b

₁

X · · · ).

Proof. See [18] IV. §9 Theorem 9.2.

3.2 The group F (µ _M )

Let K be a d-dimensional local field containing the local field K. For example,

we may consider K to be K{{T

₁

}} · · · {{T

_d−1

}}. Denote by F (µ

K

) the group with

underlying set µ

K

and operation defined by the formal group F . More generally, if

M is an algebraic extension of K we define

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