R ESEARCH I NSTITUTEFOR M ATHEMATICAL S CIENCESKYOTOUNIVERSITY,Kyoto,Japan ByYuichiroHOSHINovember2012 AnoteonthegeometricityofopenhomomorphismsbetweentheabsoluteGaloisgroupsofp-adiclocalfields RIMS-1765

A note on the geometricity of open homomorphisms between the absolute Galois groups of p-adic local fields

By

Yuichiro HOSHI

November 2012

R _ESEARCH I NSTITUTE FOR M ATHEMATICAL S _CIENCES

KYOTO UNIVERSITY, Kyoto, Japan

BETWEEN THE ABSOLUTE GALOIS GROUPS OFp-ADIC LOCAL FIELDS

YUICHIRO HOSHI NOVEMBER 2012

ABSTRACT. In the present paper, we prove that an open continuous homomor- phism between the absolute Galois groups ofp-adic local fields isgeometric [i.e., roughly speaking, arises from an embedding of fields] if and only if the homomorphism isHT-preserving[i.e., roughly speaking, satisfies the condition that the pull-back by the homomorphism of every Hodge-Tate representation is Hodge-Tate].

CONTENTS

Introduction 1

1. HT-preserving homomorphisms 3

2. Injectivity result 5

3. The main results 8

References 12

INTRODUCTION

Let pbe a prime number. WriteQp for thep-adic completion of the field of rational numbersQ. For∈ {◦,•}, let k be ap-adic local field [i.e., a finite extension ofQp] andk an algebraic closure ofk. WriteGk

def= Gal(k/k).

Let

α:G_k◦ −→G_k•

be an open continuous homomorphism. In [1], [2], S. Mochizuki discussed the geometricity[cf. [2], Definition 3.1, (iv)] of such anα. In particular, Mochizuki proved that the following conditions are equivalent [cf. [2], Theorem 3.5, (i)]:

(i) αisgeometric, i.e., arises from an isomorphism of fieldsk_• →^∼ k_◦that determines an embeddingk_•,→k_◦.

(ii) αisof CHT-type[cf. [2], Definition 3.1, (iv)], i.e.,αiscompatiblewith the respectivep-adiccyclotomic charactersofGk◦,Gk•, and, moreover, there exists an isomorphism of topological modules [butnot necessarily the topological fields]k^∧_◦ →^∼ k_•^∧— where, for∈ {◦,•}, we writek^∧for the p-adic completion of k — that is compatiblewith the respective natural actions ofG_k◦,G_k• onk^∧_◦,k^∧_• [relative toα].

2010Mathematics Subject Classification. Primary 11S20; Secondary 11S31.

This research was supported by Grant-in-Aid for Scientific Research (C), No. 24540016, Japan Society for the Promotion of Science.

1

(iii) αis of 01-qLT-type [cf. [2], Definition 3.1, (iv)], i.e., for every pair of open subgroupsH_◦⊆Gk_◦,H_• ⊆Gk_• ofGk_◦,Gk_• such thatα(H_◦)⊆H_•, and every character φ: H_• → E^× of qLT-type [cf. [2], Definition 3.1, (iii)] — whereE is ap-adic local field all of whoseQp-conjugates are contained in the fixed fieldsk^H_◦^◦,k^H_•^• — the compositeH_◦ ^α|→^H◦ H_• →^φ E^×isHodge-Tate, and theset of Hodge-Tate weights of this composite is contained in{0,1}.

We shall say that α isHT-preserving [cf. Definition 1.3, (i)] ifαpreserves the Hodge-Tate-ness ofp-adic representations, i.e., for every finite dimensional continuous representationφ:Gk• →GLn(Qp)ofGk•, ifφis Hodge-Tate, then the compositeG_k◦ →^α G_k• →^φ GL_n(Qp)is Hodge-Tate. Then it is immediate that

ifαisof CHT-type, thenαisHT-preserving.

Moreover, since a characterof qLT-typeisHodge-Tate, and its set of Hodge-Tate weights iscontainedin{0,1}, one verifies easily that

if α is not only HT-preserving but also preserves the sets of Hodge-Tate weightsof Hodge-Tate representations, thenαis of 01-qLT-type.

On the other hand, it does not seem to be clear that the following assertion holds:

If αisHT-preserving, then αis either of CHT-typeor of 01- qLT-type.

In particular, the following question may be regarded as a natural question concerning the geometricityof open continuous homomorphisms between the absolute Galois groups ofp-adic local fields:

Is every HT-preserving open continuous homomorphism be- tween the absolute Galois groups ofp-adic local fieldsgeomet- ric?

In the present paper, we answer this question in the affirmative by refining the argument of Mochizuki applied in [1], [2]. The main consequence of the present paper is as follows [cf. Corollaries 3.4; 3.5].

Theorem. Letpbe a prime number. For∈ {◦,•}, letkbe ap-adic local field andkan algebraic closure ofk. WriteG_k

def= Gal(k/k). Let

α:Gk_◦ −→Gk_•

be an open continuous homomorphism. Thenαisgeometric[cf.[2], Definition 3.1,(iv)]if and only ifαisHT-preserving[cf. Definition 1.3,(i)]. In particular, if we write

Emb(k_•/k_•, k_◦/k_◦)

for the set of isomorphisms of fieldsk_• →^∼ k_◦that determine embeddingsk_• ,→ k_◦;

Emb(k•, k◦) for the set of embeddings of fieldsk_•,→k_◦;

Hom^open_HT (Gk◦, Gk•)

for the set of HT-preserving open continuous homomorphisms G_k◦ → G_k•, then we have a commutative diagram of natural maps

Emb(k•/k•, k◦/k◦) −−−−→^∼ Hom^open_HT (Gk◦, Gk•)



 y



 y

Emb(k_•, k_◦) −−−−→^∼ Hom^open_HT (G_k◦, G_k•)/Inn(G_k•)

— where the vertical arrows are surjective, and the horizontal arrows are bijective.

1. HT-PRESERVING HOMOMORPHISMS

In the present §1, we define the notion of anHT-preserving [i.e., “Hodge- Tate-preserving”] homomorphism [cf. Definition 1.3, (i), below]. Let p be a prime number. WriteQ^pfor thep-adic completion of the field of rational num- bersQ. For∈ {◦,•,∅}, letk be ap-adic local field [i.e., a finite extension of Qp] andkan algebraic closure ofk. Writeok for the ring of integers ofk, Gk

def= Gal(k/k),Ik ⊆Gk for the inertia subgroup ofGk, andPk ⊆Ik

for the wild inertia subgroup of Gk. Now let us recall from local class field theorythat we have a natural isomorphism

G^ab_k −→^∼ (k^×)^∧

— where we write (k^×)^∧ for the profinite completion of the topological group k^×— that determines an isomorphism

(G^ab_k ⊇) Im(Ik,→GkG_k^ab)−→^∼ o^×_k (⊆(k^×)^∧).

In the following, let us regardo^×_k as a closed subgroup ofG^ab_k by means of this isomorphism, i.e.,o^×_k ⊆G^ab_k .

Proposition 1.1. Letα:G_k◦ → G_k• be an open continuous homomorphism.

Thenα(I_k◦),α(P_k◦)⊆G_k• areopensubgroups ofI_k•,P_k•, respectively. More- over, it holds thatKer(α)⊆P_k◦.

Proof. This follows immediately from [2], Proposition 3.4 [cf. also the proof of

[2], Proposition 3.4].

Definition 1.2.

(i) Let A be a topological group; φ1, φ2: Gk → A continuous homomor- phisms. Then we shall say thatφ1 isinertially equivalenttoφ2 ifφ1

and φ₂ coincide on an open subgroup of I_k ⊆ G_k [cf. the discussion preceding [4], Chapter III,§A.5, Theorem 2].

(ii) Let E be a finite Galois extension ofQp that admits an embedding σ:E ,→k. Letπ∈o_k be a uniformizer ofo_k. Then we shall write

χ^LT_σ,π:Gk−→E^×

for the continuous character obtained by forming the composite G_kG^ab_k →^∼ (k^×)^{∧ ∼}→o^×_k ×Zbo^×_k →o^×_E →^∼ o^×_E,→E^×

— where the first arrow is the natural surjection, the second arrow is the natural isomorphism arising from local class field theory, the third arrow is the isomorphism determined by the uniformizerπ∈ok, the fourth arrow is the first projection, the fifth arrow is the homo- morphism induced by the norm mapk^× → E^× [with respect to the embeddingσ], the sixth arrow is the isomorphism given by mappinga toa⁻¹, and the seventh arrow is the natural inclusion [cf. [4], Chapter

III,§A.4]. SinceI_k⊆G_ksurjects ontoo_k× {1} ⊆o_k×Zb[cf. the discus- sion at the beginning of§1], one verifies easily that theinertial equiva- lence class[cf. (i)] ofχ^LT_σ,πdoesnot dependon the choice ofπ∈ok. Thus, we shall often writeχ^LT_σ to denoteχ^LT_σ,π for some unspecified choice of π∈ok.

Definition 1.3. Letα:G_k◦ →G_k• be an open continuous homomorphism.

(i) We shall say that α is HT-preserving [i.e., “Hodge-Tate-preserving”]

if, for every finite dimensional continuous representation φ: Gk• → GLn(Qp)ofGk_•that is Hodge-Tate, the compositeGk_◦

→α Gk_•

→φ GLn(Qp) is Hodge-Tate.

(ii) We shall say that αisof HT-qLT-type [i.e., “Hodge-Tate-quasi-Lubin- Tate” type] (respectively, of weakly HT-qLT-type[i.e., “weakly Hodge- Tate-quasi-Lubin-Tate” type]) if, for

• every pair of respective finite extensionsk_◦⁰ (⊆k◦),k_•⁰ (⊆k•)ofk◦, k•such thatα(Gk⁰_◦)⊆Gk⁰_•,

• every finite Galois extensionEofQpthat admits a pair of embed- dingsσ_◦:E ,→k⁰_◦,σ_•:E ,→k⁰_•,

the composite

Gk⁰_◦ α|Gk0

−→◦ Gk⁰_• χ^LT_σ•

−→E^×

[cf. Definition 1.2, (ii)] is Hodge-Tate (respectively, is inertially equiv- alent [cf. Definition 1.2, (i)] to a continuous characterGk⁰_◦ →E^×that factors through the natural open injection Gk⁰_◦ ,→ Gal(k_◦/E) deter- mined by the embeddingsE ,^σ→^◦ k⁰_◦ ,→k_◦) [cf. Proposition 1.1]. [Here, we note that, as is well-known — cf., e.g., [4], Chapter III,§A.1, Corol- lary 2 — the issue of whether or not a finite dimensional continuous representation isHodge-Tate depends onlyon theinertial equivalence classof the given representation.]

Lemma 1.4. Letα:Gk_◦ → Gk_• be an open continuous homomorphism. Con- sider the following four conditions:

(1) αisHT-preserving[cf. Definition 1.3,(i)].

(1⁰) For every pair of respective finite extensionsk_◦⁰ (⊆k_◦),k_•⁰ (⊆k_•)of k_◦, k_• such that α(Gk_◦⁰) ⊆ Gk⁰_•, the restriction α|G_k0

◦: Gk⁰_◦ → Gk⁰_• is HT- preserving.

(2) αisof HT-qLT-type[cf. Definition 1.3,(ii)].

(3) αisof weakly HT-qLT-type[cf. Definition 1.3,(ii)].

Then we have an equivalence and implications (1)⇐⇒(1⁰) =⇒(2) =⇒(3).

Proof. The implication (1⁰)⇒(1) is immediate. First, we verify the implication (1)⇒(1⁰). Letk_◦⁰ (⊆k_◦),k⁰_•(⊆k_•)be respective finite extensions ofk_◦,k_•such thatα(Gk⁰_◦)⊆Gk⁰_•; φ: Gk⁰_• →GLn(Qp)a finite dimensional continuous repre- sentation ofGk• that isHodge-Tate. Now let us observe [cf., e.g., [4], Chapter III,§A.1, Corollary 2] that, to verify that the compositeφ◦α|G_k0

◦ isHodge-Tate

— by replacing k⁰_◦, k_•⁰ by suitable finite extensions of k⁰_◦, k_•⁰, respectively — we may assume without loss of generality that k⁰_◦, k⁰_• are Galois overk_◦, k_•, respectively. Writeφ_k• for the finite dimensional continuous representation of G_k• obtained by inducing φ from G_k⁰_• to G_k•. Then since [one verifies easily that]φ_k•|_Gk0

• is isomorphic to the direct product of[k⁰_•:k_•]copies ofφ, it holds that φk• isHodge-Tate. Thus, sinceαisHT-preserving, it holds thatφk• ◦α,

hence also(φ_k• ◦α)|_Gk0

◦, isHodge-Tate. On the other hand, one verifies easily thatφ◦α|G_k0

◦ is isomorphic to a subrepresentation of(φk• ◦α)|G_k0

◦. In partic- ular, we conclude thatφ◦α|G_k0

◦ isHodge-Tate. This completes the proof of the implication (1)⇒(1⁰).

The implication (1⁰)⇒(2) follows from the fact that “χ^LT_σ,π” defined in Defi- nition 1.2, (ii), isHodge-Tate[cf. [4], Chapter III,§A.5, Corollary]. Finally, we verify the implication (2)⇒(3). We shall apply the notational conventions es- tablished in Definition 1.3, (ii). Then sinceαisof HT-qLT-type, the character χ: G_k⁰_◦ →E^×obtained by forming the composite

G_k⁰_◦

α|Gk0

−→◦ G_k⁰_• ^χ

LT

−→σ• E^×

isHodge-Tate. Thus, sinceEisGaloisoverQp, it follows immediately from [4], Chapter III, §A.5, Corollary, thatχisinertially equivalent [cf. Definition 1.2, (i)] to the character

Y

σ∈Gal(E/Qp)

(χ^LT_σ◦◦σ)^nσ:G_k⁰_◦ −→E^×

for some choices of integersnσ. On the other hand, one verifies easily fromlocal class field theory that this character isinertially equivalentto the restriction toG_k0

◦ ⊆Gal(k_◦/E)of the character Y

σ∈Gal(E/Qp)

(χ^LT_σ )^nσ: Gal(k◦/E)−→E^×.

This completes the proof of the implication (2)⇒(3), hence also of Lemma 1.4.

Remark 1.4.1. In the notation of Lemma 1.4, consider the following four con- ditions:

(4) αisof qLT-type[cf. [2], Definition 3.1, (iv)].

(5) αisof 01-qLT-type[cf. [2], Definition 3.1, (iv)].

(6) αisof CHT-type[cf. [2], Definition 3.1, (iv)].

(7) αisof HT-type[cf. [2], Definition 3.1, (iv)].

Then we have equivalences and implications

(7)⇐= (4)⇐⇒(5)⇐⇒(6) (=⇒(1)⇐⇒(1⁰) =⇒(2) =⇒(3)).

Indeed, the equivalences (4)⇔(5)⇔(6) follow from [2], Theorem 3.5, (i); the implications (6)⇒(1) and (6)⇒(7) are immediate. If, moreover,αisinjective, then we have equivalences and implications

(4)⇐⇒(5)⇐⇒(6)⇐⇒(7) (=⇒(1)⇐⇒(1⁰) =⇒(2) =⇒(3)).

Indeed, the implication (7)⇒(6) follows immediately from [1], Proposition 1.1.

2. INJECTIVITY RESULT

In the present §2, we prove that every open continuous homomorphismof weakly HT-qLT-type isinjective [cf. Proposition 2.4 below]. We maintain the notation of the preceding§1.

Definition 2.1.

(i) LetGbe a profinite group. Then we shall write (G) G^p-ab-free

for the maximal pro-pabelian torsion-free quotient ofG.

(ii) Let A be an abelian topological group and φ: G_k → A a continuous homomorphism. Then we shall write

iner-dim(φ)^def= dim_Qp(φ(I_k)^p-ab-free⊗_ZpQp)

[cf. (i)] and refer toiner-dim(φ)as theinertial dimensionofφ.

Lemma 2.2. LetAbe an abelian topological group andφ:G_k →Aa continu- ous homomorphism. Then the following hold:

(i) It holds that

0≤iner-dim(φ)≤[k:Qp] [cf. Definition 2.1,(ii)].

(ii) LetH ⊆I_k be a closed subgroup ofI_k. Suppose thatH contains an open subgroup ofPk [e.g.,H is an open subgroup ofIk orPk]. Then

iner-dim(φ) = dim_Qp(φ(H)^p-ab-free⊗_ZpQ^p) [cf. Definition 2.1,(i)].

(iii) Let φ⁰: G_k → A be a continuous homomorphism that is inertially equivalenttoφ[cf. Definition 1.2,(i)]. Then

iner-dim(φ) = iner-dim(φ⁰). (iv) In the notation of Definition 1.2,(ii), it holds that

iner-dim(χ^LT_σ ) = [E:Qp] [cf.(iii)].

(v) Letα: Gk◦ →Gkbe anopencontinuous homomorphism. Then it holds that

iner-dim(φ) = iner-dim(φ◦α). Proof. First, I claim that the following assertion holds:

Claim 2.2.A: The natural surjectionI_k φ(I_k)^p-ab-free factors through the natural surjectionI_k o^×_k (o^×_k)^p-ab-free [cf. the discussion at the beginning of§1].

Indeed, this follows immediately from our assumption thatAisabelian. This completes the proof of Claim 2.2.A.

Assertion (i) follows immediately from Claim 2.2.A, together with the fact that(o^×_k)^p-ab-free⊗_ZpQp is of dimension[k :Qp]. Assertion (ii) follows immedi- ately from Claim 2.2.A, together with the [easily verified] fact that the compos- ite Pk ,→ Ik o^×_k isopen. Assertion (iii) follows immediately from assertion (ii). Assertion (iv) follows immediately from the definition of the characterχ^LT_σ , together with the fact that(o^×_E)^p-ab-free⊗_ZpQp is of dimension[E:Qp]. Finally, we verify assertion (v). Let us first observe that it follows from Proposition 1.1 that αdetermines anopenhomomorphismPk◦ →Pk. Thus, assertion (v) fol- lows immediately from assertion (ii). This completes the proof of assertion

(v).

Lemma 2.3. LetN⊆G_kbe anontrivialnormal closed subgroup ofG_k. Then there exists an open subgroupH ⊆G_kofG_ksuch that the image of the composite N∩H ,→H H^p-ab-free[cf. Definition 2.1,(i)]isnontrivial.

Proof. Assume that, for every open subgroup H ⊆Gk ofGk, the image of the composite N∩H ,→ H H^p-ab-free istrivial, i.e., if we writeJH ⊆H for the kernel of the natural surjectionH H^p-ab-free, thenN∩H ⊆JH. Now sinceN isnontrivial, it is immediate that there exists a normal open subgroupH ⊆Gk

such that the composite N ,→ Gk Gk/H is nontrivial. In particular, one verifies easily that, to verify Lemma 2.3, by replacingGkby the inverse image

of the image of N in G_k/H via G_k G_k/H, we may assume without loss of generality that the compositeN ,→Gk Gk/H is [nontrivial and]surjective.

Thus, since [we have assumed that] N∩H ⊆JH, it follows immediately that the compositeN ,→Gk Gk/JH determines asplittingof the exact sequence of profinite groups

1−→H^p-ab-free−→Gk/JH −→Gk/H−→1.

[Here, we note that since H ⊆ G_k isnormal, and J_H ⊆ H ischaracteristic, one verifies easily that J_H is normal inG_k.] In particular, since N ⊆ G_k is normal, the natural action [determined by the above exact sequence] ofGk/H onH^p-ab-free, hence also onH^p-ab-free⊗_ZpQp, istrivial. On the other hand, if we writek⁰(⊆k)for the finite Galois extension ofkcorresponding toH ⊆Gk, then it follows immediately from local class field theory that there exists aGk/H (= Gal(k⁰/k))-equivariant injection ofQp-vector spacesk⁰ ,→ H^p-ab-free⊗_ZpQp, whichcontradictsthe fact that the action ofG_k/HonH^p-ab-free⊗_ZpQpistrivial.

This completes the proof of Lemma 2.3.

Proposition 2.4. Letα:Gk◦ → Gk• be an open continuous homomorphism.

Suppose that αisof weakly HT-qLT-type[cf. Definition 1.3, (ii)]. Thenαis injective.

Proof. Assume that the homomorphism α is not injective. Then it follows immediately from Lemma 2.3 that there exists a finite Galois extension E of Qp that admits a pair of embeddings E ,→ k◦, E ,→ k• such that if we write E◦ ⊆ k◦, E• ⊆ k• for the respective images of these embeddings [so E_◦ ←^∼ E →^∼ E_•], thenk_◦ ⊆E_◦,k_• ⊆E_•, and, moreover, the image of the com- positeKer(α)∩G_E◦ ,→G_E◦ G^p-ab-free_E

◦ [cf. Definition 2.1, (i)] isnontrivial.

Let k⁰_◦(⊆ k_◦)be a finite extension ofk_◦ such thatE_◦ ⊆ k_◦⁰, and, moreover, α(Gk⁰_◦)⊆GE•. Writeχfor the composite

Gk⁰_◦ α|Gk0

−→◦ GE_• χ^LT_id

−→E_•^× (←^∼ E^{× ∼}→E_◦^×) [cf. Definition 1.2, (ii)]. Then since α|G_k0

◦ isopen, it follows from Lemma 2.2, (iv), (v), that

iner-dim(χ) = iner-dim(χ^LT_id ) = [E•:Qp]

[cf. Definition 2.1, (ii)]. On the other hand, since αisof weakly HT-qLT-type, the characterχ isinertially equivalentto the continuous character factors as the composite

Gk_◦⁰ −→GE_◦ χ_E◦

−→E_◦^× (←^∼ E^{× ∼}→E_•^×)

of the natural open injectionGk_◦⁰ ,→GE◦and a continuous characterχE◦:GE◦ → E_◦^×. Thus, it follows from Lemma 2.2, (iii), (v), that

([E_•:Qp] =) iner-dim(χ) = iner-dim(χE◦).

Now let us recall from Proposition 1.1 thatKer(α) ⊆P_k◦. In particular, it holds thatKer(α) = Ker(α)∩Ik◦, which thus implies thatKer(α)∩Ik⁰_◦ isopenin Ker(α). On the other hand, it follows from the definition ofχthatKer(α)∩I_k◦⁰ (= Ker(α)∩G_k⁰_◦)⊆ Ker(χ). Thus, since χ isinertially equivalent toχ_E◦|_Gk0

◦, we conclude that there exists anopensubgroupJ ⊆Ker(α)ofKer(α)such that J ⊆Ker(χE◦)⊆ GE◦. Now sinceJ ⊆Ker(α)isopeninKer(α), and [we have assumed that] the image of the composite Ker(α)∩GE◦ ,→ GE◦ G^p-ab-free_E◦ isnontrivial, it follows that the image of the compositeJ ,→ G_E◦ G^p-ab-free_E

◦

is nontrivial. Thus, one verifies easily that the image of the homomorphism J → o^×_E◦ (⊆ G^ab_E◦) [cf. the discussion at the beginning of §1] determined by

the compositeJ ,→G_E◦ G^ab_E

◦ [where we recall thatJ ⊆ I_E◦] isinfinite. In particular, since J ⊆ Ker(χE◦), we conclude that the kernel of the character (IE◦ )o^×_E◦ →E^×_◦ determined by the restriction ofχE◦ toIE◦ ⊆GE◦ isinfinite.

Thus, we obtain an inequality

([E_•:Qp] =) iner-dim(χE◦)<dim_Qp((o^×_E

◦)^p-ab-free⊗_ZpQp) = [E_◦:Qp], which contradicts the fact that E_◦ ←^∼ E →^∼ E_•. This completes the proof of

Proposition 2.4.

3. THE MAIN RESULTS

In the present§3, we prove the main theorem of the present paper [cf. The- orem 3.3 below]. We maintain the notation of§1.

Definition 3.1. Letα:Gk◦

→∼ Gk• be a continuousisomorphismandβ: k_• →^∼ k◦ an isomorphism of fields. Then we shall say thatβ isinertially compatible withαif the composite

o^×_k

• ,→k_•^×→^∼ k^×_◦ ,→(k^×_◦)^∧

— where the second arrow is the isomorphism determined by β — and the composite

o^×_k

• ,→G^ab_k• →^∼ G^ab_k◦ →^∼ (k^×_◦)^∧

— where the first arrow is the natural inclusion arising from local class field theory [cf. the discussion at the beginning of §1], the second arrow is the iso- morphism determined byα⁻¹, and the third arrow is the isomorphism arising from local class field theory — coincide on an open subgroup ofo^×_k

•. Lemma 3.2. Letα:Gk◦

→∼ Gk• be a continuous isomorphism;β1,β2:k_• →^∼ k_◦ isomorphisms of fields. Suppose thatβ1,β2areinertially compatiblewithα [cf. Definition 3.1]. Thenβ1=β2.

Proof. Sinceβ1,β2areinertially compatiblewithα, one verifies easily from the various definitions involved that there exists an open subgroupS•⊆o^×_k• ofo^×_k• such thatβ₁|S• =β₂|S•. On the other hand, let us recall from [1], Lemma 4.1, that the sub-Qp-vector space ofk_•generated byS_•coincideswithk_•. Thus, the equalityβ₁|_S• =β₂|_S• implies the equalityβ₁=β₂. This completes the proof of

Lemma 3.2.

Theorem 3.3. Letpbe a prime number. For∈ {◦,•}, letkbe ap-adic local field andkan algebraic closure ofk. WriteGk

def= Gal(k/k). Let

α:G_k◦ −→G_k•

be an open continuous homomorphism. Suppose thatαisof HT-qLT-type[cf.

Definition 1.3,(ii)]. Thenαisgeometric[cf.[2], Definition 3.1,(iv)], i.e., arises from an isomorphism of fieldsk_•→^∼ k_◦that determines an embeddingk_•,→k_◦. Proof. First, let us observe that it follows from Proposition 2.4, together with the implication (2)⇒(3) of Lemma 1.4, thatαisinjective. Next, let us observe that, to verify Theorem 3.3, by replacingGk• by the image ofα, we may assume without loss of generality thatαis anisomorphism.

The following argument is essentially the same as the argument applied in [1] to prove the main theorem of [1]. Now I claim that the following assertion holds:

Claim 3.3.A: Suppose that k_◦ isGalois overQp. Then there exists a(n) [necessarilyunique— cf. Lemma 3.2] isomorphism of fieldsβk•,k◦:k• ∼

→k◦that isinertially compatiblewithα[cf.

Definition 3.1].

Indeed, let Ebe a finite Galois extension ofQ^p that admits embeddingsE ,→ k_◦,E ,→ k_• such that if we writeE_◦ ⊆k_◦,E_• ⊆k_• for the respective images of these embeddings [soE_◦←^∼ E →^∼ E_•], thenk_◦ ⊆E_◦,k_• ⊆E_•. Letk_◦⁰ (⊆k_◦) be a finite Galois extension ofk◦such that k_◦⁰ containsE◦, and, moreover, the finite [necessarily Galois] extensionk⁰_• (⊆k_•)of k_• corresponding to the open subgroupα(G_k◦⁰)⊆G_k• containsE_•. For∈ {◦,•}, writeσ:E,→k⁰for the natural inclusion. Writeχfor the composite

Gk⁰_◦ α|Gk0

∼◦

−→ Gk⁰_• χ^LT_σ•

−→E^×_• (←^∼ E^{× ∼}→E^×_◦).

Then sinceαisof HT-qLT-type, it holds thatχ isHodge-Tate. Thus, sinceE_◦ is Galois over Qp, it follows from [4], Chapter III,§A.5, Corollary, thatχ is inertially equivalentto the character

Y

σ∈Gal(E◦/Qp)

(χ^LT_σ◦◦σ)^nσ:Gk⁰_◦ −→E_◦^× (←^∼ E^{× ∼}→E_•^×)

for some choices of integersnσ. For∈ {◦,•}, writeVer_k⁰

/k: G^ab_k

→G^ab_k0

for theVerlagerung mapwith re- spect to the finite Galois extensionk⁰ /k. Then sinceχisinertially equivalent toQ

σ∈Gal(E◦/Qp)(χ^LT_σ◦◦σ)^nσ, and [one verifies easily fromlocal class field theory that] Ver_k⁰

/k maps o^×_k

⊆ G^ab_k

[cf. the discussion at the beginning of §1] to o^×_k0

⊆G^ab_k0

, we conclude that there exists an open subgroupS_◦⊆o^×_k

◦ (⊆G^ab_k◦)of o^×_k

◦ such that if we writeS_•⊆o^×_k

• for the image ofS_◦⊆o^×_k

◦ by the isomorphism (G^ab_k◦ ⊇) o^×_k

◦

−→∼ o^×_k

• (⊆G^ab_k•)

induced byα[where let us recall from Proposition 1.1 that αinduces an iso- morphismIk◦

→∼ Ik•], then the diagram of topological modules

S_◦ −−−−→ G^ab_k

◦

Ver_k0

◦/k◦

−−−−−−→ G^ab_k0

◦

Q

σ∈Gal(E◦/Qp)(χ^LT_{σ◦ ◦σ})^nσ

−−−−−−−−−−−−−−−−→ E_◦^× ←−−−−^∼ E^×

o



 y

S_• −−−−→ G^ab_k

•

Ver_k0

•/k•

−−−−−−→ G^ab_k0

•

χ^LT_σ•

−−−−→ E_•^× ←−−−−^∼ E^×

— where the left-hand vertical arrow is the isomorphism induced byα, and the left-hand horizontal arrows are the natural inclusions — commutes. On the other hand, it follows immediately fromlocal class field theory, together with Definition 1.2, (ii), that, for∈ {◦,•}, if we writeIm(Ik)⊆G^ab_k

for the image of the compositeIk,→Gk G^ab_k

[i.e., “o^×_k

”⊆G^ab_k

— cf. the discussion at the beginning of§1], then we have commutative diagrams of topological modules

Im(Ik◦)

Ver_k0

◦/k◦

−−−−−−→ Im(Ik⁰_◦)

Q

σ∈Gal(E◦/Qp)(χ^LT_{σ◦ ◦σ})^nσ

−−−−−−−−−−−−−−−−→ E^×_◦ ←^∼ E^×

o





y ^o



 y

o^×_k

◦ −−−−→ o^×_k0

◦

Q

σ∈Gal(E◦/Qp)(σ⁻¹◦Nm_k0

◦/E◦)^nσ

−−−−−−−−−−−−−−−−−−−−−−→ E_◦^×←^∼ E^×,

Im(Ik•)

Ver_k0

•/k•

−−−−−−→ Im(Ik⁰_•) ^χ

LT

−−−−→σ• E_•^×←^∼ E^×

o





y ^o



 y

o^×_k• −−−−→ o^×_k0

•

Nm_k0

•/E•

−−−−−−→ E_•^×←^∼ E^×

— where the left-hand and middle vertical arrows are isomorphisms that arise fromlocal class field theory; the lower left-hand horizontal arrows are the ho- momorphisms induced by the natural inclusions k_◦ ,→ k⁰_◦, k_• ,→ k_•⁰, respec- tively; we write “Nm” for the norm map. In particular, if, for ∈ {◦,•}, we writeIm(S)⊆E^×

for the image ofSinE^×

, then the following hold:

(a) Sincek_◦⊆E_◦ ⊆k_◦⁰, andk_◦isGaloisoverQp [which thus implies that everyσ∈Gal(E_◦/Qp)preservesk_◦⊆E_◦], it holds that

Im(S◦) = Y

σ∈Gal(E◦/Qp)

(σ⁻¹◦Nmk⁰_◦/E◦)(S◦)^nσ = Y

σ∈Gal(E◦/Qp)

σ⁻¹(S^nσ·[k

0

◦:E_◦]

◦ )⊆k_◦^×,

i.e., that the subgroupIm(S_◦)⊆E_◦^×iscontainedink^×_◦ ⊆E_◦^×.

(b) Since k_• ⊆ E_• ⊆ k_•⁰, it holds that the subgroup Im(S_•) ⊆ E_•^× coin- cideswith the subgroup(o^×_k

•)^[k0•:E•] ⊆E_•^×, which thus implies that the subgroupIm(S•)⊆E_•^×is anopensubgroup ofo^×_k• ⊆E_•^×.

For each∈ {◦,•}, writeV⊆Efor the sub-Qp-vector space ofEgenerated byIm(S)⊆E. Now we have a commutative diagram of topological modules

Im(S_◦) −−−−→ E_◦^× ←−−−−^∼ E^×

o



 y

Im(S_•) −−−−→ E_•^× ←−−−−^∼ E^×

— where the left-hand vertical arrow is the isomorphism induced by α, and the left-hand horizontal arrows are the natural inclusions. Thus, it is immedi- ate that the isomorphisms of fieldsE_• ←^∼ E →^∼ E_◦ determine an isomorphism V_• →^∼ V_◦, which thus implies thatdim_Qp(V_◦) = dim_Qp(V_•). Moreover, it follows from (a) (respectively, (b), together with [1], Lemma 4.1) that V_◦ ⊆ k_◦ ⊆ E_◦ (respectively,V_•=k_•⊆E_•). Thus, since[k_◦:Qp] = [k_•:Qp][cf. [1], Proposition 1.2], we conclude thatV◦=k◦,V•=k•, and, moreover, the isomorphism ofQp- vector spacesV_•→^∼ V_◦[determined by theisomorphisms of fieldsE_•←^∼ E→^∼ E_◦] iscompatiblewith the structures of fields ofk_◦,k_•. In particular, we obtain an isomorphism of fieldsβk•,k◦:k_•=V_• →^∼ V_◦ =k_◦. On the other hand, it follows from the definition of βk•,k◦, together with the above discussion concerning Im(S), thatβk•,k◦ isinertially compatiblewithα. This completes the proof of Claim 3.3.A.

Next, I claim that the following assertion holds:

Claim 3.3.B: For every pair of respective finite extensionsk_◦⁰ (⊆k◦), k⁰_• (⊆k•)of k◦,k• such thatα(Gk⁰_◦) = Gk⁰_•, there ex- ists a(n) [necessarilyunique — cf. Lemma 3.2] isomorphism of fieldsβk_•⁰,k⁰_◦: k_•⁰ →^∼ k_◦⁰ that isinertially compatiblewith the restrictionα|G_k0

◦:Gk_◦⁰

→∼ Gk⁰_•.

Indeed, letk⁰⁰_◦ (⊆k_◦)be a finite extension ofk_◦⁰ that isGaloisoverQp. Write k_•⁰⁰ (⊆ k_•)for the finite [necessarily Galois] extension of k_•⁰ corresponding to the open subgroup α(G_k⁰⁰_◦) ⊆ G_k•. Then it follows from Claim 3.3.A that there exists an isomorphism of fields βk⁰⁰_•,k⁰⁰_◦: k⁰⁰_• →^∼ k⁰⁰_◦ that is inertially com- patible with the restrictionα|G_k00

◦ :Gk⁰⁰_◦

→∼ Gk_•⁰⁰. Then one verifies easily from

Lemma 3.2, together with the fact thatβ_k⁰⁰_•,k⁰⁰_◦ isinertially compatiblewith the restrictionα|G_k00

◦, thatβk_•⁰⁰,k⁰⁰_◦ iscompatiblewith the respective natural actions of Gal(k⁰⁰_◦/k_◦⁰),Gal(k_•⁰⁰/k_•⁰)onk_◦⁰⁰,k⁰⁰_• [relative to the isomorphismGal(k_◦⁰⁰/k⁰_◦) = Gk⁰_◦/Gk⁰⁰_◦

→∼ Gk⁰_•/Gk⁰⁰_• = Gal(k_•⁰⁰/k⁰_•)induced byα|G_k0

◦]. Thus, we conclude that the isomorphism βk⁰⁰_•,k_◦⁰⁰ determines an isomorphism βk⁰_•,k⁰_◦:k⁰_• →^∼ k_◦⁰. On the other hand, again by Lemma 3.2, together with the fact thatβ_k⁰⁰_•,k⁰⁰_◦ isinertially compatiblewith the restrictionα|G_k00

◦, it follows immediately that this isomor- phismβk⁰_•,k⁰_◦ isinertially compatiblewith the restrictionα|G_k0

◦. This completes the proof of Claim 3.3.B.

Now, by applying Claim 3.3.B to the various finite extensions ofk◦, we obtain an isomorphism of fieldsβ_k

•,k◦:k_•→^∼ k_◦that determines an isomorphismk_• →^∼ k_◦. Moreover, again by applying Claim 3.3.B, one verifies easily thatαarises from this isomorphismβ_k

•,k◦. This completes the proof of Theorem 3.3 Remark 3.3.1. Theorem 3.3 leads naturally to the following observation:

Letpbe anoddprime number andQpan algebraic closure of the p-adic completion Qp of the field of rational numbersQ. WriteG_Qp^def= Gal(Qp/Qp). Then there exist an automorphism αof G_Qp and a finite dimensional continuous representation φ:G_Qp → GL_n(Qp) of G_Qp such that φ is potentially locally algebraic, i.e., the restriction ofφto an open subgroup ofG_Qp islocally algebraic[cf. [4], Chapter III,§1, Definition] [hence Hodge-Tate], the set of Hodge-Tate weights ofφiscontained in{0,1}, butφ◦αisnot Hodge-Tate.

Indeed, let us first observe that it follows immediately from the discussion given at the final part of [3], Chapter VII,§5, that we have an automorphism αofG_Qpthat isnot geometric[cf. [2], Definition 3.1, (iv)]. Thus, it follows from Theorem 3.3 that αisnot of HT-qLT-type[cf. Definition 1.3, (ii)]. In particu- lar, since the character “χ^LT_σ ” defined in Definition 1.2, (ii), islocally algebraic [cf. [4], Chapter III, §1, Example (2)], and the set of Hodge-Tate weights is containedin{0,1}[cf., e.g., [4], Chapter III,§A.5, Theorem 2], it follows from the definition of a homomorphismof HT-qLT-typethat there exist normal open subgroups H1, H2 ⊆G_Qp and a finite dimensional continuous representation φH₂: H2 → GLn(Qp) of H2 such that α(H1) ⊆ H2, φH₂ is locally algebraic, the set of Hodge-Tate weights of φH₂ is contained in {0,1}, and, moreover, φH2 ◦α:H1 →GLn(Qp)isnot Hodge-Tate. Thus, it follows immediately from a similar argument to the argument applied in the proof of the implication (1)

⇒ (1⁰) of Lemma 1.4 that if we writeφfor the finite dimensional continuous representation ofG_Qp obtained by inducingφ_H2 fromH₂toG_Qp, thenφispo- tentially locally algebraic[cf. also [4], Chapter III,§A.7, Theorem 3], the set of Hodge-Tate weights ofφiscontainedin{0,1}, butφ◦αisnot Hodge-Tate.

Corollary 3.4. In the notation of Theorem 3.3, consider the following nine conditions:

(1) αisHT-preserving[cf. Definition 1.3,(i)].

(2) αisof HT-qLT-type[cf. Definition 1.3,(ii)].

(3) αisgeometric[cf.[2], Definition 3.1,(iv)].

(4) αisof qLT-type[cf.[2], Definition 3.1,(iv)].

(5) αisof 01-qLT-type[cf.[2], Definition 3.1,(iv)].

(6) αisof CHT-type[cf.[2], Definition 3.1,(iv)].

(7) αisof HT-type[cf.[2], Definition 3.1,(iv)].

(8) αis[an isomorphism and]RF-preserving[cf.[2], Definition 3.6,(iii)].

(9) α is [an isomorphism and] uniformly toral [cf. [2], Definition 3.6, (iii)].

Then we have equivalences and implications

(8)⇐⇒(9) =⇒(1)⇐⇒(2)⇐⇒(3)⇐⇒(4)⇐⇒(5)⇐⇒(6) =⇒(7). If, moreover,αis anisomorphism, then the above nine conditions areequiv- alent.

Proof. Let us recall from Remark 1.4.1 that we have implications (4) =⇒(5) =⇒(6) =⇒(1) =⇒(2)and(6) =⇒(7).

The implication (2)⇒(3) follows from Theorem 3.3. The implication (3)⇒(4) follows from [2], Theorem 3.5, (i). The equivalence (8)⇔(9) and the implica- tion (8) ⇒(3) follow from [2], Corollary 3.7. Finally, the implication (7)⇒(6) (respectively, (3)⇒(8)) in the case whereαis anisomorphismfollows immedi- ately from [1], Proposition 1.1 (respectively, [2], Corollary 3.7). This completes

the proof of Corollary 3.4.

Corollary 3.5. Letpbe a prime number. For∈ {◦,•}, letkbe ap-adic local field andkan algebraic closure ofk. WriteGk

def= Gal(k/k);Emb(k_•/k_•, k_◦/k_◦) for the set of isomorphisms of fieldsk_• →^∼ k_◦that determine embeddingsk_• ,→ k_◦; Emb(k_•, k_◦)for the set of embeddings of fieldsk_• ,→ k_◦;Hom^open_HT (Gk_◦, Gk_•) for the set of open continuous homomorphisms α: Gk_◦ → Gk_• that are HT- preserving[cf. Definition 1.3,(i)], i.e., for every finite dimensional continuous representation φ: Gk• → GLn(Qp) of Gk•, if φ is Hodge-Tate, then φ◦α is Hodge-Tate. Then we have a commutative diagram of natural maps

Emb(k•/k•, k◦/k◦) −−−−→^∼ Hom^open_HT (Gk◦, Gk•)



 y



 y

Emb(k_•, k_◦) −−−−→^∼ Hom^open_HT (G_k◦, G_k•)/Inn(G_k•)

— where the vertical arrows are surjective, and the horizontal arrows are bijective.

Proof. Theinjectivityof the horizontal arrows follow immediately from thein- jectivityportion of [1], Theorem 4.2 [cf. also the proof of [1], Theorem 4.2]. The surjectivityof the horizontal arrows follow immediately from Theorem 3.3, to- gether with the implication (1)⇒(2) of Lemma 1.4. This completes the proof

of Corollary 3.5.

REFERENCES

[1] S. Mochizuki, A version of the Grothendieck conjecture for p-adic local fields,Internat. J.

Math.8(1997), 499–506.

[2] S. Mochizuki, Topics in absolute anabelian geometry I: Generalities,J. Math. Sci. Univ. Tokyo.

19(2012), 139–242.

[3] J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of number fields, Second edition, Grundlehren der Mathematischen Wissenschaften,323, Springer-Verlag, Berlin, 2008.

[4] J. P. Serre,Abelianl-adic representations and elliptic curves, McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute W. A. Benjamin, Inc., New York-Amsterdam 1968.

(Yuichiro Hoshi) RESEARCH INSTITUTE FORMATHEMATICAL SCIENCES, KYOTOUNIVER- SITY, KYOTO606-8502, JAPAN

E-mail address:[email protected]