Valuations in number fields

2.6 Valuations in number fields

References: [Neu99]

Now, we are going to particularize some of the previous results to the case of number fields.

Definition 2.6.1. A prime (or place) of a number field K is a class of equivalent valuations.

The primes corresponding to archimedian valuations are called infinite primes. Since a valu-ation of K gives, by restriction, a valuvalu-ation of Q, the infinite primes of a number field correspond to the extensions of the unique archimedian valuation | · |_∞ of Q. Consequently, by Proposi-tion 2.5.1, these primes are represented by valuaProposi-tions of the form |x| = |τ x|_∞ where now | · |_∞ represents the classical absolute value of C and τ is a Q-embedding of K into C. Also by Propo-sition 2.5.1, two embeddings τ and τ⁰ give the same valuation if and only if they are conjugate.

If τ (K) ⊆ R, we say that the corresponding prime is called a real prime. Otherwise, it is called a complex prime. Among the [K : Q] Q-embeddings of K into C, we get a real prime from each real embedding and a complex prime from each pair of conjugate complex embeddings.

The primes corresponding to non-archimedian valuations are called finite primes, and they must extend non-archimedian valuations of Q, which are, up to equivalence, the p-adic valuations.

If

(p) = P^e₁¹. . . P^e_r^r, then, for each Pi, the exponential valuation

1 ei

ordP_i,

where ordP(a) is defined as the exponent of P in the factorization of the fractional ideal (a) into prime ideals, clearly extends ordp to K. Moreover, the ramification index and inertial degree of the corresponding valuations agree with the ramification index and inertial degree defined for classical prime ideals, so that Proposition 2.5.4 implies that these are all the valuations extending ordp.

To each prime p of K, finite or infinite, we assign a normalized valuation. For an infinite prime p|∞, the normalized valuation is given by:

|x|p = |τ x|∞, if p is real, and by

|x|p = |τ x|²_∞,

if p is complex, where in both cases τ is a Q-embedding giving rise to the prime p. Actually, the definition in the case of a complex prime does not meet the definition of a valuation, as it does not satisfy the triangle inequality, but it is anyway convenient to define the normalized valuation in this way. For a finite prime p|p, the normalized valuation is given by:

|x|_p= Nm(p)^−ordp(x)= p^−fpordp(x)

For infinite primes, we define the residue class field as the completion K_p, so that for a extension of number fields L/K and P|p the inertial degree is

f_P|p= [L_p: K_p] ,

which can only be 1 or 2, and the ramification index e_P|pis always taken to be one. With these definitions, it is easy to see that Proposition 2.5.4 also holds for infinite primes.

68 CHAPTER 2. VALUATIONS AND LOCAL FIELDS

Proposition 2.6.2. (Product formula) Let K be a number field. Then, for any x ∈ K^×, Y

p

|x|p= 1 ,

where p runs over all primes of K (both finite and infinite).

Proof. We begin by proving the formula in the case K = Q. In this case, given x ∈ Q,

Now, let K be any number field, and take any x ∈ K. For the infinite primes we have Y

p|∞

|x|p= Y

σ∈Gal(K/Q)

|σ| = |Nm_K/Q(x)|∞.

For the finite primes dividing a finite prime p of Q, we have Y

Chapter 3

Local class field theory

In this chapter K will always denote a non-archimedian local field and L an algebraic extension of K. All extensions will be considered within the same algebraic closure of K. We use the notation K^sfor the separable closure of K. The groups of units of K and L are denoted by U_K and U_L; the maximal ideals by p_K and p_L, and the residue class fields by κ and λ, respectively.

The notations ord_K and ord_L will refer to the normalized exponential valuations of K and L, respectively. We also define U_K^(m)= 1 + p^m_K and U_L^(m)= 1 + p^m_L for all m > 1. Local uniformizing parameters will also be referred to as prime elements.

3.1 The cohomology of unramified extensions

References: [Mil13]

In this section the extension L/K will always be unramified. Set G = Gal(L/K).

Assume that L/K is finite. Let π be a local uniformizing parameter of K. Since L/K is unramified, π is also a local uniformizing parameter of L. Also because L/K is unramified, G ' Gal(λ/κ), where the isomorphism is the natural one. This fact allows us to regard λ and λ^× as G-modules. Therefore, the isomorphisms

U_L/U_L⁽¹⁾→ λ^× a 7→ [a]

and

U^(m)/U^(m+1)→ λ 1 + aπ^m7→ [a]

from Proposition 2.1.10 are in fact G-isomorphisms.

Lemma 3.1.1. Assume that L/K is finite. Then H_T^r(G, λ) = 0 and H_T^r(G, λ^×) = 0 for all r ∈ Z.

Proof. Since G ' Gal(λ/κ) is cyclic, by Proposition 1.9.8 we need only prove the result for r = 1 and r = 2. In fact, since λ and λ^× are finite groups, by Proposition 1.9.11 the corresponding Herbrand quotients are both 1, so that it is enough to prove the result for r = 1. For λ this is a consequence of Proposition 1.8.6 and for λ^× it is a consequence of Hilbert’s theorem 90 (Proposition 1.8.5).

69

70 CHAPTER 3. LOCAL CLASS FIELD THEORY Remark 19. Note that, in particular, for r = 0, we get κ/Tr_λ/κλ = H_T⁰(G, λ) = 0 and κ^×/Nm_λ/κλ^× = H_T⁰(G, λ^×) = 0, so that the trace map Tr_λ/κ : λ → κ and the norm map Nm_λ/κ: λ^×→ κ^× are surjective.

Proposition 3.1.2. Assume that L/K is finite. Then the map NmL/K : UL→ UKis surjective.

Proof. It is straightforward that the diagram

UL λ^×

UK κ^×

Nm_L/K Nm_λ/κ

and the diagrams

U_L^(m) λ^×

U_K^(m) κ^×

Nm_L/K Tr_λ/κ

for m > 1, where the horizontal arrows are the maps from the proof of Proposition 2.1.10, are commutative. Observe that the horizontal arrows and the right vertical arrows are surjective.

Therefore, given u ∈ U_K, there exists some v₀∈ U_L which is mapped through the first diagram to the same element in κ^× as u. Therefore, we have u/Nm_L/Kv₀∈ U_K⁽¹⁾. Now, in the same way and using the corresponding diagram, we can obtain an element v1 ∈ U_L⁽¹⁾ for which we have u/ Nm_L/K(v0) · Nm_L/K(v1)
= u/NmL/K(v0v1) ∈ U_K⁽²⁾. Continuing in this way, we obtain a sequence (v_n)_n with v_n ∈ U_L⁽ⁿ⁾ such that u/Nm_L/K(v₀· · · vn) ∈ U_K⁽ⁿ⁺¹⁾ for all n > 0. Since vn ∈ U_L⁽ⁿ⁾ for all n > 1, it is easy to see that the sequenceQn

i=0vi converges. Let v =Q∞ i=0vi. Since the norm map is continuous (because K is complete, so that all σ ∈ G preserve the valuation of L), we have Nm_L/K(v) = limnNm_L/K(v0· · · vn), which equals u because of the property u/Nm_L/K(v₀· · · v_n) ∈ U_K⁽ⁿ⁺¹⁾for all n > 0.

Proposition 3.1.3. Assume that L/K is finite. Then H_T^r(G, UL) = 0 for all r ∈ Z.

Proof. Since G is cyclic, it suffices to prove the result for r = 0 and r = 1. For r = 0, it is a consequence of the previous proposition.

Now, observe that

L^×= U_L× π^Z. Since π ∈ K, it remains fixed by G, so that we have

L^×' UL× Z

as G-modules, where we are considering, as usual, the trivial action on Z. Therefore, H¹(G, L^×) ' H¹(G, U_L) × H¹(G, Z) ;

but, by Hilbert’s theorem 90, H¹(G, L^×) = 0, so that we have H¹(G, UL) = 0.

Corollary 3.1.4. We have H^r(G, U_L) = 0 for all r > 0 (now L/K can be infinite).

3.1. THE COHOMOLOGY OF UNRAMIFIED EXTENSIONS 71

Proof. For all r > 0, we have

H^r(G, U_L) = lim

−→H^r(G/H, U_L^H) ,

where the direct limit is taken over the open subgroups H of G of finite index. This is equivalent to taking the direct limit over the finite subextensions F of L, i.e.

H^r(G, UL) = lim

−→H^r(Gal(F/K), UF) , and the desired result follows from the finite case.

We have just seen that H^r(G, U_L) = 0 for all r > 0. Therefore, from the short exact sequence

0 UL L^{× ordL} Z 0

we get isomorphisms

H^r(G, L^×)−^'→ H^r(G, Z) for all r > 0 (and, in particular, for r = 2).

On the other hand, since H^r(G, Q) = 0 for all r > 0 (see Lemma 1.9.7), the short exact sequence

0 Z Q Q/Z 0

, gives isomorphisms

H^r(G, Q/Z)−^'→ H^r+1(G, Z) for all r > 0 (and, in particular for r = 1).

For any Galois extension of fields E/F , we will write H²(E/F ) for H²(Gal(E/F ), E^×).

Definition 3.1.5. The invariant map for the extension L/K inv_L/K : H²(L/K) → Q/Z is the map obtained from the composite

H²(G, L^×)−^'→ H²(G, Z)−^'→ H¹(G, Q/Z) ' Homcts(G, Q/Z)−−−−−−−−−−→ Q/Z ,^{ϕ7→ϕ(FrobL/K)} where the first two isomorphisms are the ones which we had just obtained.

Remark 20. Since the Fr¨obenius element FrobL/K generates G, any ϕ ∈ Homcts(G, Q/Z) is de-termined by its action on FrobL/K, so that the last map in the definition of the invariant map is injective and, consequently, the invariant map is itself injective (the other arrows are isomor-phisms). If L is a finite unramified extension, then, since G is cyclic, a map ϕ ∈ Hom(G, Q/Z) can map the Fr¨obenius element to any element in _[L:K]¹ Z/Z, so that we get an isomorphism

invL/K : H²(L/K)−^'→ 1

[L : K]Z/Z .

72 CHAPTER 3. LOCAL CLASS FIELD THEORY Lemma 3.1.6. The invariant maps are compatible in the sense that, if L and M are unramified extensions of K and L ⊆ M , then the diagram

H²(L/K) Q/Z

H²(M/K) Q/Z

Inf

inv_L/K

inv_M/K

=

is commutative.

Proof. Thinking of the cohomology groups in terms of cochains, it is straghtforward to check that the inflation map is compatible with all the homomorphisms in the definition of the invariant maps.

Proposition 3.1.7. The map

inv_Kun/K : H²(K^un/K) → Q/Z is an isomorphism.

Proof. We have already seen that the invariant maps are injective, so that we need only prove surjectivity. To that end, observe that, by the previous lemma, for every finite unramified extension L/K we have the commutative diagram

H²(L/K) _[L:K]¹ Z/Z

H²(K^un/K) Q/Z

Inf

inv_L/K

inv_{Kun /K}

,

where the first arrow is an isomorphism. Since K is a local field, there exist finite unramified extensions of all finite degree (recall that the finite unramified extensions of K are in bijection with the finite extensions of κ, which is a finite field), whereby surjectivity follows.

Remark 21. Since

H²(K^un/K) ' lim

−→H²(F/K) ,

where the direct limit is taken over all finite subextensions of K^un/K and the homomorphisms defining the direct limit are precisely the inflation maps, the map inv_Kun/Kis uniquely determined by the maps invF /K defined for finite unramified extensions F .

Until the end of this section, we will use the simplified notation invK to denote the map inv_Kun/K.

Assume that L/K is finite. The element of H²(L/K) which is mapped to _[L:K]¹ ∈ _[L:K]¹ Z/Z by the isomorphism invL/K will be referred to as the local fundamental class for the extension L/K and will be denoted by u_L/K.

The G-module L^× satisfies the hypothesis of Tate’s theorem: for every subgroup H of G we have that H¹(H, L^×) = 0 because of Hilbert’s theorem 90, and H²(H, L^×) = H²(L/L^H) is cyclic of order |H| because of the isomorphism

inv_L/LH : H²(L/L^H) → 1

[L : L^H]Z/Z .

3.1. THE COHOMOLOGY OF UNRAMIFIED EXTENSIONS 73 Therefore, by Tate’s theorem, cup-product with the fundamental class u_L/K provides isomor-phisms

H_T^r(G, Z) → HT^r+2(G, L^×)

for all r ∈ Z. In particular, for r = −2 and taking into account the isomorphism G = G^ab' H²(G, Z)

from Proposition 1.9.2, we get an isomorphism

γ_L/K : G → K^×/Nm_L/KL^×.

Lemma 3.1.8. Assume that L/K is finite. Let σ = Frob_L/K, let π be a local uniformizing parameter of K and let n = [L : K]. Then, the fundamental class u_L/K is represented by the cochain ϕ defined by

ϕ(σⁱ, σ^j) =

(1 if i + j 6 n − 1 π if i + j > n − 1 .

Proof. The preimage of ¹_n ∈ ¹_nZ/Z by the last map from the definition of the invariant map inv_L/K is the unique χ ∈ Hom(G, Q/Z) ' H¹(G, Q/Z) such that χ(σ) =_n¹, i.e. it is the element of H¹(G, Q/Z) represented by the 1-cochain defined by

χ(σⁱ) = i n.

It is straightforward to perform the calculation of the image of this element by the connecting map

H¹(G, Q/Z) → H²(G, Z) coming from the short exact sequence

0 Z Q Q/Z 0

, which yields the 2-cochain defined by

ϕ(σⁱ, σ^j) =

(0 if i + j 6 n − 1 1 if i + j > n − 1 .

Finally, taking into account that π remains fixed by G and ord_L(π) = 1, we obtain the desired result.

Proposition 3.1.9. With the previous definitions and assumptions, the map γ_L/K maps the Fr¨obenius element σ to the class of prime elements in K^×Nm_L^×.

Proof. From the proof of Tate’s theorem (Theorem 1.9.14), the isomorphism H_T⁻²(G, Z) → HT⁰(G, L^×)

is obtained as the composite of the connecting maps

H_T⁻²(G, Z) → HT⁻¹(G, I_G)

74 CHAPTER 3. LOCAL CLASS FIELD THEORY and

H_T⁻¹(G, IG) → H_T⁰(G, L^×) coming from the short exact sequences

0 I_G Z[G] Z 0

and

0 L^× L^×(ϕ) IG 0

,

respectively. The isomorphism G ' H⁻²(G, Z) is obtained through de composite of the first of the previous connecting maps and the homomorphism

H_T⁻¹(G, I_G) = I_G/I_G² → G^ab= G g − 1 + I_G² 7→ g

(see Proposition 1.9.2). Therefore, it is clear that the Fr¨obenius element σ is mapped to σ −1+I_G² in I_G/I_G².

The second connecting map comes from applying the snake lemma to the diagram L^×_G L^×(ϕ)G (IG)G 0

0 L^×G L^×(ϕ)^G I_G^G

Nm_G Nm_G Nm_G

,

as it was defined through the diagram (1.9). In particular, it is the map going from the kernel of the last vertical arrow to the cokernel of the first vertical arrow. The group H_T⁻¹(G, I_G) = I_G/I_G² coincides with (I_G)_G, and a preimage of σ − 1 + I_G² in L^×(ϕ)_G = L^×(ϕ)/I_GL^×(ϕ) is given by x_σ+ I_GL^×(ϕ). For i = 0, 1, . . . , n − 1, we have

σⁱxσ= x_σi+1− x_σi+ ϕ(σ, σⁱ) , where xid= ϕ(1, 1) = 1. Hence,

NmGxσ=

n−1

Y

i=0

ϕ(σ, σⁱ) = π , whereby the desired result follows.

In document Class field theory (Page 69-76)