On a ramification bound of torsion semi-stable representations over a local field

Journal of Number Theory 129 (2009) 2474–2503

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Journal of Number Theory www.elsevier.com/locate/jnt

On a ramification bound of torsion semi-stable representations over a local field Shin Hattori ∗ Faculty of Mathematics, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan

a r t i c l e

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Article history: Received 8 September 2008 Revised 9 April 2009 Available online 4 July 2009 Communicated by David Goss

Let p be a rational prime, k be a perfect field of characteristic p, W = W (k) be the ring of Witt vectors, K be a finite totally ramified extension of Frac( W ) of degree e and r be a non-negative integer satisfying r < p − 1. In this paper, we prove the upper ( j)

numbering ramification group G K for j > u ( K , r , n) acts trivially on the pn -torsion semi-stable G K -representations with Hodge–Tate weights in {0, . . . , r }, where u ( K , 0, n) = 0, u ( K , 1, n) = 1 + e (n + 1/( p − 1)) and u ( K , r , n) = 1 − p −n + e (n + r /( p − 1)) for 1 < r < p − 1. © 2009 Elsevier Inc. All rights reserved.

Keywords: Ramification Integral p-adic Hodge theory Semi-stable representation

1. Introduction Let p be a rational prime, k be a perfect field of characteristic p, W = W (k) be the ring of Witt vectors and K be a finite totally ramified extension of K 0 = Frac( W ) of degree e = e ( K ). We normalize the valuation v K of K as v K ( p ) = e and extend this to any algebraic closure of K . Let the maximal ideal of K be denoted by m K , an algebraic closure of K by K¯ and the absolute Galois group of K by ( j) G K = Gal( K¯ / K ). Let G K denote the j-th upper numbering ramification group in the sense of [10]. ( j)

j −1

Namely, we put G K = G K , where the latter is the upper numbering ramification group defined in [18]. Consider a proper smooth scheme X K over K and put X K¯ = X K × K K¯ . Let L ⊇ L be G K -stable Z p -lattices in the r-th etale cohomology group H er´ t ( X K¯ , Q p ) such that the quotient L/L is killed ( j)

by pn . In [10], Fontaine conjectured the upper numbering ramification group G K acts trivially on the G K -modules L/L and H er´ t ( X K¯ , Z/ pn Z) for j > e (n + r /( p − 1)) if X K has good reduction. For

*

Fax: +81 92 642 4198. E-mail address: [email protected]. URL: http://www2.math.kyushu-u.ac.jp/~shin-h/.

0022-314X/$ – see front matter doi:10.1016/j.jnt.2009.04.012

©

2009 Elsevier Inc. All rights reserved.

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e = 1 and r < p − 1, this conjecture was proved independently by himself ([11], for n = 1) and Abrashkin ([3], for any n), using the theory of Fontaine and Laffaille [13] and the comparison theorem of Fontaine and Messing ([14], see also [5,7]) between the etale cohomology groups of X K and the crystalline cohomology groups of the reduction of X K . From these results, they also showed some rareness of a proper smooth scheme over Q with everywhere good reduction [11, Théorème 1], [2, Section 7]. In fact, they proved this ramification bound for the torsion crystalline representations of G K with Hodge–Tate weights in {0, . . . , r } in the case where K is absolutely unramified. On the other hand, for a torsion semi-stable representation with Hodge–Tate weights in the same range, a similar ramification bound for e = 1 and n = 1 is obtained by Breuil (see [7, Proposition 9.2.2.2]). He showed, assuming the Griffiths transversality which in general does not hold, that ( j) if e = 1 and r < p − 1, then the ramification group G K acts trivially on the mod p semi-stable representations for j > 2 + 1/( p − 1). In this paper, we prove a ramification bound for the torsion semi-stable representations of G K with Hodge–Tate weights in {0, . . . , r } with no assumption on e but under the assumption r < p − 1. Let π be a uniformizer of K , E (u ) ∈ W [u ] be the Eisenstein polynomial of π over W and S be the p-adic completion of the divided power envelope of W [u ] with respect to the ideal ( E (u )). Consider r ,φ, N a category Mod/ S ∞ of filtered (φr , N )-modules over the ring S and a G K -module ∗ ˆ T st ,π (M) = Hom S ,Filr ,φr , N (M, A st,∞ )

ˆ st,∞ is a p-adic period ring [6]. Then our main theorem is the following. for M ∈ Mod/ S ∞ , where A r ,φ, N

r ,φ, N

Theorem 1.1. Let r be a non-negative integer such that r < p − 1. Let M be an object of the category Mod/ S ∞ ( j)

which is killed by pn . Then the j-th upper numbering ramification group G K acts trivially on the G K -module ∗ (M) for j > u ( K , r , n), where T st ,π

⎧ ⎨0

1 u ( K , r , n) = 1 + e (n + p −1 ) ⎩ 1 r 1 − pn + e (n + p − ) 1

(r = 0), (r = 1), (1 < r < p − 1).

We can check that this bound is sharp for r  1 (Remark 5.15). From this theorem and [10, Proposition 1.3], we have the following corollary. Corollary 1.2. Let the notation be as in the theorem and L be the finite extension of K cut out by the G K -module ∗ (M). Namely, the finite extension L is defined by T st ,π







∗ G L = Ker G K → Aut T st ,π (M) .

Let D L / K denote the different of the extension L / K . Then we have the inequality

v K (D L / K ) < u ( K , r , n) for r > 0 and v K (D L / K ) = 0 for r = 0. Combining these results with a theorem of Liu [17, Theorem 2.3.5] or a theorem of Caruso [8, Théorème 1.1], we will show the corollary below. Corollary 1.3. Let r be a non-negative integer such that r < p − 1. Then the same bounds as in Theorem 1.1 and Corollary 1.2 are also valid for the torsion G K -modules of the following two cases:

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1. The G K -module L/L , where L ⊇ L are G K -stable Z p -lattices in a semi-stable p-adic representation V with Hodge–Tate weights in {0, . . . , r } such that L/L is killed by pn . 2. The G K -module H er´ t ( X K¯ , Z/ pn Z), where X K is a proper smooth algebraic variety over K which has a proper semi-stable model over O K and r satisfies er < p − 1 for n = 1 and e (r + 1) < p − 1 for n > 1. For the proof of Theorem 1.1, we basically follow a beautiful argument of Abrashkin [3]. We may assume p  3 and r  1. Consider the finite Galois extension

Fn = K



n

π 1 / p , ζ p n +1



of K whose upper ramification is bounded by u ( K , r , n). Let L n be the finite Galois extension of F n cut ∗ (M)| out by T st G F n . Then we bound the ramification of L n over K . For this, we show that to study ,π this G F n -module we can use a variant over a smaller coefficient ring Σ of filtered (φr , N )-modules over S. In precise, we set

q y Σ = W u , E (u ) p / p . This ring Σ is small enough for the method of Abrashkin, in which he uses filtered modules of Fontaine and Laffaille [13] whose coefficient ring is W , to work also in the case where K is absolutely ramified. 2. Filtered (φr , N )-modules of Breuil In this section, we recall the theory of filtered (φr , N )-modules over S of Breuil, which is developed by himself and most recently by Caruso and Liu (see for example [6,8,17,9]). In what follows, we always take the divided power envelope of a W -algebra with the compatibility condition with the natural divided power structure on pW . Let p be a rational prime and σ be the Frobenius endomorphism of W . We fix once and for all a uniformizer π of K and a system {πn }n∈Z0 of p-power roots of π in K¯ such that π0 = π and πn =

πnp+1 for any n. Let E (u ) be the Eisenstein polynomial of π over W and set S = ( W [u ]PD )∧ , where PD

means the divided power envelope and this is taken with respect to the ideal ( E (u )), and ∧ means the p-adic completion. The ring S is endowed with the σ -semilinear endomorphism φ : u → u p and a natural filtration Filt S induced by the divided power structure such that φ(Filt S ) ⊆ pt S for 0  t  p − 1. We set φt = p −t φ|Filt S and c = φ1 ( E (u )) ∈ S × . Let N denote the W -linear derivation on S defined by the formula N (u ) = −u. We also define a filtration, φ , φt and N on S n = S / pn S similarly. r ,φ, N to be the category consisting of the following Let r ∈ {0, . . . , p − 2} be an integer. Set  Mod/ S data:

• an S-module M and its S-submodule Filr M containing Filr S · M, • a φ -semilinear map φr : Filr M → M satisfying

φr (sr m) = φr (sr )φ(m) for any sr ∈ Filr S and m ∈ M, where we set φ(m) = c −r φr ( E (u )r m), • a W -linear map N : M → M such that – N (sm) = N (s)m + sN (m) for any s ∈ S and m ∈ M, – E (u ) N (Filr M) ⊆ Filr M, – the following diagram is commutative:

Filr M

φr

E (u ) N

Filr M

M cN

φr

M,

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r ,φ, N and the morphisms of  Mod/ S are defined to be the S-linear maps preserving Filr and commuting r ,φ with φr and N. The category defined in the same way but dropping the data N is denoted by  Mod/ S . r ,φ, N

These categories have obvious notions of exact sequences. Let Mod/ S 1  Modr ,φ, N consisting of /S

denote the full subcategory of

M such that M is free of finite rank over S 1 and generated as an S 1 -module r ,φ, N

r ,φ, N

by the image of φr . We write Mod/ S ∞ for the smallest full subcategory which contains Mod/ S 1 r ,φ, N

is stable under extensions. We let Mod/ S

and

denote the full subcategory consisting of M such that

• the S-module M is free of finite rank and generated by the image of φr , • the quotient M/ Filr M is p-torsion free. r ,φ

r ,φ

r ,φ

We define full subcategories Mod/ S 1 , Mod/ S ∞ and Mod/ S

ˆ ∈ Modr ,φ, N (resp. M /S r ,φ (resp. Mod/ S ∞ ).

r ,φ Mod/ S ),

r ,φ of  Mod/ S in a similar way. For

ˆ / pn M ˆ has a natural structure as an object of Mod the quotient M /S∞

r ,φ, N

For p-torsion objects, we also have the following categories. Consider the k-algebra k[u ]/(u ep ) ∼ = S 1 / Fil p S 1 and let this algebra be denoted by S˜ 1 . The algebra S˜ 1 is equipped with the natural filtrar ,φ, N tion, φ and N induced by those of S. Namely, Filt S˜ 1 = u et S˜ 1 , φ(u ) = u p and N (u ) = −u. Let  Mod ˜ /S1

denote the category consisting of the following data:

˜ and its S˜ 1 -submodule Filr M ˜ containing u er M ˜, • an S˜ 1 -module M ˜ →M ˜, • a φ -semilinear map φr : Filr M ˜ such that ˜ →M • a k-linear map N : M ˜, – N (sm) = N (s)m + sN (m) for any s ∈ S˜ 1 and m ∈ M ˜ ) ⊆ Filr M ˜, – u e N (Filr M – the following diagram is commutative:

˜ Filr M

φr

˜ M

ue N

cN

˜ Filr M

φr

˜, M r ,φ, N

and whose morphisms are defined as before. Its full subcategory Mod ˜ /S1 condition:

is defined by the following

˜ is free of finite rank and generated by the image of φr . • as an S˜ 1 -module, M r ,φ r ,φ We define categories  Mod ˜ and Mod ˜ similarly. Then we can show as in the proof of [4, Propo/S1 /S1 sition 2.2.2.1] that the natural functor M → M/ Fil p S · M induces equivalences of categories r ,φ, N r ,φ, N r ,φ r ,φ and T 0 : Mod/ S 1 → Mod ˜ . T : Mod/ S 1 → Mod ˜

/S1 φ

/S1

For r = 0, let Mod/ W ∞ be the category consisting of the following data:

˜ • a finite torsion W -module M, ˜ ˜ → M. • a σ -semilinear automorphism φ : M Let

κ be the kernel of the natural surjection S → W defined by u → 0. Since Tor1S (M, S /κ S ) = 0 0,φ

for any M ∈ Mod/ S ∞ , the proofs of [8, Lemme 2.2.7, Proposition 2.2.8] work also for the category 0,φ, N

Mod/ S ∞

and we have a commutative diagram of categories

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0,φ, N

0,φ

Mod/ S ∞

Mod/ S ∞

φ

Mod/ W ∞ , where the downward arrows and horizontal arrow are defined by M → M/κ M and forgetting N respectively and these three arrows are equivalences of categories. ˆ st be p-adic period rings. These are constructed as follows. Put O˜ ¯ = O ¯ / p O ¯ . Let A crys and A K K K Set R to be the ring

R = lim (O˜ K¯ ← O˜ K¯ ← · · ·), ←− where every arrow is the p-power map. For an element x = (xi )i ∈Z0 ∈ R and an integer n  0, we set pm

x(n) = lim xˆ n+m ∈ OC , m→∞ where xˆ i is a lift of xi in O K¯ and OC is the p-adic completion of O K¯ . Let v p denote the valuation of OC normalized as v p ( p ) = 1. Then the ring R is a complete valuation ring whose valuation of an element x ∈ R is given by v R (x) = v p (x(0) ). We define a natural ring homomorphism θ by

θ : W ( R ) → OC ,  (n) (x0 , x1 , . . .) → pn xn . n0

Then A crys is the p-adic completion of the divided power envelope of W ( R ) with respect to the prin-

ˆ st is the p-adic completion of the divided power polynomial ring A crys X  cipal ideal Ker(θ) and A

ˆ st,∞ = Aˆ st ⊗W K 0 / W . Put over A crys . We set A crys,∞ = A crys ⊗ W K 0 / W and A where we abusively let

π = (πn )n∈Z0 ∈ R, πn denote the image of πn ∈ O K¯ in O˜ K¯ . These rings are considered

ˆ st and Aˆ st → A crys which are defined by u → as S-algebras by the ring homomorphisms S → A [π ]/(1 + X ) and X → 0, respectively. The ring A crys is endowed with a natural filtration induced by the divided power structure, a natural Frobenius endomorphism φ and the φ -semilinear map r ,φ φt = p −t φ|Filt A crys . With these structures, A crys and A crys,∞ are considered as objects of  Mod/ S . Moreˆ st , its filtration is defined over, the absolute Galois group G K acts naturally on these two rings. As for A by

ˆ st = Filt A

 i 0

ai

X i i!

ai ∈ Filt −i A crys , lim ai = 0

ˆ st by and the Frobenius structure of A crys extends to A

φ( X ) = (1 + X ) p − 1, φt = p −t φ|Filt Aˆ st .

i →∞



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ˆ st defined by N ( X ) = 1 + X . The rings Aˆ st and We write N also for the A crys -linear derivation on A ˆ st,∞ are objects of  Modr ,φ, N . The G K -action on A crys naturally extends to an action on Aˆ st . Indeed, A /S

ˆ st is defined by the formula the action of g ∈ G K on A g( X ) =





ε ( g ) (1 + X ) − 1,

where g (πn ) = εn ( g )πn and ε ( g ) = (εn ( g ))n∈Z0 ∈ R with the abusive notation as above. These rings have other descriptions, as follows. For an integer n  1, put W n = W / pn W and let ˜ ¯ . We define a W n -algebra structure W n (O˜ K¯ ) be the ring of Witt vectors of length n associated to O K

on W n (O˜ K¯ ) by twisting the natural W n -algebra structure by phism

σ −n . Then the natural ring homomor-

θn : W n (O˜ K¯ ) → O K¯ / pn O K¯ , (a0 , . . . , an−1 ) →

n −1 

p n −i

p i aˆ i

,

i =0

where aˆ i is a lift of ai in O K¯ , is W n -linear. Let us denote W nPD (O˜ K¯ ) the divided power envelope of W n (O˜ K¯ ) with respect to the ideal Ker(θn ). This ring is considered as an S-algebra by u → [πn ]. This ring also has a natural filtration defined by the divided power structure, and a natural G K -module structure. The Frobenius endomorphism of the ring of Witt vectors induces on this ring a φ -semilinear Frobenius endomorphism, which is denoted also by φ . Then, by the S-linear transition maps PD ˜ ˜ W nPD +1 (O K¯ ) → W n (O K¯ ),

 p p  (a0 , . . . , an ) → a0 , . . . , an−1 , these S-algebras form a projective system compatible with all the structures. Using this transition map, a φ -semilinear map

φr : Filr W nPD (O˜ K¯ ) → W nPD (O˜ K¯ ) ˜ is defined by setting φr (x) to be the image of p −r φ(ˆx), where xˆ is a lift of x in Filr W nPD +r (O K¯ ). By PD ˜ definition, the maps φr are also compatible with the transition maps. The S-algebra W n (O K¯ ) is r ,φ r ,φ considered as an object of  Mod/ S . Then we have a natural isomorphism in  Mod/ S

A crys / pn A crys → W nPD (O˜ K¯ ),

(x0 , . . . , xn−1 ) → (x0,n , . . . , xn−1,n ), where we set xi = (xi ,k )k∈Z0 for xi ∈ R.

Similarly, the divided power polynomial ring W nPD (O˜ K¯ ) X  over W nPD (O˜ K¯ ) is considered as an S-algebra by u → [πn ]/(1 + X ). This ring has a natural filtration coming from the divided power structure. We define a G K -action on this ring by

g( X ) =





εn ( g ) (1 + X ) − 1.

We also define a φ -semilinear Frobenius endomorphism, which we also write as φ , by φ( X ) = (1 + X ) p − 1 and a W nPD (O˜ K¯ )-linear derivation N by N ( X ) = 1 + X . These rings form a projective

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system of S-algebras compatible with all the structures by the transition maps defined by the maps above and X → X . We define φ -semilinear maps

φr : Filr W nPD (O˜ K¯ ) X  → W nPD (O˜ K¯ ) X  compatible with the transition maps as before. The S-algebra W nPD (O˜ K¯ ) X  is considered as an object

r ,φ, N r ,φ, N and there exists a natural isomorphism in  Mod/ S of  Mod/ S

ˆ st / pn Aˆ st → W nPD (O˜ ¯ ) X , A K (x0 , . . . , xn−1 ) → (x0,n , . . . , xn−1,n ), X → X which is G K -linear.

r ,φ, N ∗ (M) to be Put K n = K (πn ) and K ∞ = n K n . For M ∈ Mod/ S ∞ , we define a G K -module T st ,π ∗ ˆ T st ,π (M) = Hom S ,Filr ,φr , N (M, A st,∞ ).

When M is killed by pn , we have a natural identification of G K -modules





∗ PD ˜ T st ,π (M) = Hom S ,Filr ,φr , N M, W n (O K¯ ) X  .

Note that the G K -module on the right-hand side is independent of the choice of the natural map

πk for k > n. Since

W nPD (O˜ K¯ ) X  → W nPD (O˜ K¯ ), X → 0 is G K n -linear, we also have a G K n -linear isomorphism [6, Lemme 2.3.1.1]





∗ PD ˜ T st ,π (M)|G K n → Hom S ,Filr ,φr M, W n (O K¯ ) . ∗ (M) On the other hand, for r = 0, the proof of [8, Proposition 2.3.13] shows that the G K -module T st ,π 0,φ, N is unramified for any M ∈ Mod/ S ∞ . A variant of filtered (φr , N )-modules over S is also introduced by Breuil and Kisin, and developed also by Caruso and Liu (see for example [15–17,9]). Put S = W Ju K and let φ : S → S be the

,φ σ -semilinear Frobenius endomorphism defined by φ(u ) = u p . Let  Modr/S denote the category con-

sisting of the following data:

• an S-module M, • a φ -semilinear map M → M, which is denoted also by φ , such that the cokernel of the map 1 ⊗ φ : φ ∗ M → M, where we set φ ∗ M = S ⊗φ,S M, is killed by E (u )r , r ,φ and whose morphisms are defined as before. The full subcategory of  Mod/S consisting of M such r ,φ

r ,φ

that M is free of finite rank over S/ p S (resp. over S) is denoted by Mod/S (resp. Mod/S ). We let 1 r ,φ r ,φ Mod/S denote the smallest full subcategory which contains Mod/S and is stable under extensions, ∞ 1 as before. Then we have an exact functor ([9, Proposition 2.1.2], see also [15, Proposition 1.1.11]) r ,φ

r ,φ

MS∞ : Mod/S∞ → Mod/ S ∞ .

S. Hattori / Journal of Number Theory 129 (2009) 2474–2503

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r ,φ

For M ∈ Mod/S , the filtered φr -module M = MS∞ (M) over S is defined as follows: ∞

• M = S ⊗φ,S M,

1⊗φ

• Filr M = Ker(M −−−→ S ⊗S M → ( S / Filr S ) ⊗S M), 1⊗φ φr ⊗1 • φr : Filr M −−−→ Filr S ⊗S M −− −→ S ⊗φ,S M = M. r ,φ

r ,φ

We write MS for the functor Mod/S → Mod/ S defined similarly. 3. Filtered φr -modules over Σ r ,φ

r ,φ

In this section, we define another variant Mod/Σ∞ of the category Mod/ S ∞ over a subring Σ of the ring S, and prove that they are categorically equivalent. Let p be a rational prime and r be an integer such that 0  r < p − 1. Consider the W -algebra Σ = W Ju , Y K/( E (u ) p − pY ) as in [6, Subsection 3.2]. We regard Σ as a subring of S by the map sending Y to E (u ) p / p. Then the element c = φ1 ( E (u )) ∈ S × is contained in Σ × . We define on Σ a σ -semilinear Frobenius endomorphism φ by φ(u ) = u p and φ(Y ) = p p−1 c p . Put Filt Σ = ( E (u )t , Y ) for 0  t  p − 1 and Fil p Σ = (Y ). Then we have φ(Filt Σ) ⊆ pt Σ for 0  t  p − 1. We put φt = p −t φ|Filt Σ . We also set Σn = Σ/ pn Σ and put on this ring the natural structures induced by those of Σ . r ,φ We define a category  Mod/Σ of filtered φr -modules over Σ to be the category consisting of the following data:

• a Σ -module M and its Σ -submodule Filr M containing Filr Σ · M, • a φ -semilinear map φr : Filr M → M satisfying φr (sr m) = φr (sr )φ(m) for any sr ∈ Filr Σ and m ∈ M, where we set φ(m) = c −r φr ( E (u )r m), r ,φ and whose morphisms are defined in the same manner as  Mod/ S . This category has a natural notion r ,φ

of exact sequences. We define its full subcategory Mod/Σ1 to be the category consisting of M which is r ,φ free of finite rank and generated by the image of φr as a Σ1 -module. We also let Mod/Σ∞ denote the

r ,φ r ,φ smallest full subcategory of  Mod/Σ which contains Mod/Σ1 and is stable under extensions. Moreover, r ,φ r ,φ we define a full subcategory Mod/Σ of  Mod/Σ to be the category consisting of M such that

• the Σ -module M is free of finite rank and generated by the image of φr , • the quotient M / Filr M is p-torsion free. ˆ ∈ Modr ,φ , the quotient M ˆ / pn M ˆ is naturally considered as an object of Then we see that for M /Σ r ,φ

Mod/Σ∞ .

r ,φ r ,φ The natural ring isomorphism Σ1 / Fil p Σ1 ∼ = S˜ 1 defines a functor T 0,Σ : Mod/Σ1 → Mod ˜ by M → r ,φ

r ,φ

/S1

M / Fil p Σ1 · M. Then just as in the case of the functor T 0 : Mod/ S 1 → Mod ˜ [4, Proposition 2.2.2.1], /S1 we can show the following lemma. r ,φ

r ,φ

Lemma 3.1. The functor T 0,Σ : Mod/Σ1 → Mod ˜ is an equivalence of categories. /S 1

On the other hand, [6, Proposition 2.2.1.3] and Nakayama’s lemma show the following. r ,φ

Lemma 3.2. Let M be an object of Mod/Σ1 of rank d over Σ1 . Then there exists a basis {e 1 , . . . , ed } of M such r

that Fil M = Σ1 u e 1 ⊕ · · · ⊕ Σ1 u ed + Fil p Σ1 · M for some integers r1 , . . . , rd with 0  r i  er for any i. r1

rd

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Then we can show the following lemma just as in the proof of [6, Lemme 2.3.1.3]. Lemma 3.3. The functor

M → HomΣ,Filr ,φr ( M , A crys,∞ ) r ,φ

from Mod/Σ∞ to the category of G K ∞ -modules is exact. r ,φ

r ,φ

For M ∈ Mod/Σ1 , we can show as in the case of the category Mod/ S 1 that there is an isomorphism of G K 1 -modules









HomΣ,Filr ,φr M , (O˜ K¯ )PD → Hom S˜ ,Filr ,φ T 0,Σ ( M ), O˜ K¯ , r 1

˜ ¯ is considered as an object of  Modr ,φ by the natural isomorphism where O K / S˜ 1

(O˜ K¯ )PD / Fil p (O˜ K¯ )PD → O˜ K¯ . Thus [6, Lemme 2.3.1.2] implies the following. r ,φ

Lemma 3.4. For M ∈ Mod/Σ1 , we have





#HomΣ,Filr ,φr M , (O˜ K¯ )PD = pd , where d = dimΣ1 M. r ,φ

For the category Mod/Σ∞ , we have the following lemma. r ,φ

α1 , . . . , αd ∈ Filr M such that Filr M = Σ α1 + · · · + Σ αd + Fil Σ · M and the elements e 1 = φr (α1 ), . . . , ed = φr (αd ) form a system of generators of M.

Lemma 3.5. Let M be in Mod/Σ∞ . Then there exists p

Proof. By induction and Lemma 3.2, we may assume that there exists an exact sequence of the cater ,φ gory Mod/Σ∞

0 → M  → M → M  → 0 such that the lemma holds for M  and M  . Let α1 , . . . , αl (resp. α1 , . . . , αl ) be elements of Filr M  (resp. Filr M  ) as in the lemma. Let αl ∈ Filr M be a lift of αl . Then the elements α1 , . . . , αl , α1 , . . . , αl satisfy the condition in the lemma for M. 2 r ,φ

Corollary 3.6. Let M be an object of Mod/Σ∞ and C ∈ M d (Σ) be a matrix satisfying

(α1 , . . . , αd ) = (e 1 , . . . , ed )C r ,φ with the notation of the previous lemma. Let A be an object of  Mod/Σ . Then a Σ -linear homomorphism

f : M → A preserving Filr also commutes with φr if and only if

    φr f (e 1 , . . . , ed )C = f (e 1 ), . . . , f (ed ) .

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Proof. Suppose that the latter condition holds. Then we have φr ( f (αi )) = f (φr (αi )) for any i. We only have to check the equality φr ◦ f = f ◦ φr on Fil p Σ · M. Suppose that this equality  holds on the submodule pl+1 Fil p Σ · M. For m ∈ M, we can take m ∈ Fil p Σ · M such that E (u )r m = i si αi + m . Let s be in Fil p Σ . Then we have

 

f φr pl sm



= pl φr (s)c −r



    φ(si ) f φr (αi ) + pl φr (s)c −r f φr (m ) .

i

Since φr (Fil p Σ) ⊆ p Σ , this equals to φr ( f ( pl sm)) by assumption. Thus the lemma follows by induction. 2 Corollary 3.7. Let M and A be as above and J ⊆ Filr A be a Σ -submodule of A such that φr ( J ) ⊆ J . We can r ,φ consider the Σ -module A / J naturally as an object of  Mod/Σ . Suppose that for any x ∈ J , there exists t ∈ Z0 such that φrt (x) = 0. Then the natural homomorphism of abelian groups

HomΣ,Filr ,φr ( M , A ) → HomΣ,Filr ,φr ( M , A / J ) is an isomorphism. Proof. The proof is similar to [3, Subsection 2.2]. We consider the Σ -submodule J as an object of the r ,φ r ,φ category  Mod/Σ by putting Filr J = J . By dévissage, it is enough to show that, for any M ∈ Mod/Σ1 , we have Ext Modr ,φ ( M , J ) = 0 and the map in the corollary is an isomorphism. For the first assertion, /Σ

let

0→ J →E →M →0 r ,φ be an extension in the category  Mod/Σ . Let e i ,

αi and C be as in Corollary 3.6 such that e 1 , . . . , ed

form a basis of M. Let eˆ i ∈ E be a lift of e i ∈ M. Then we have (ˆe 1 , . . . , eˆ d )C ∈ (Filr E )⊕ d and

  φr (ˆe 1 , . . . , eˆ d )C = (ˆe 1 + δ1 , . . . , eˆ d + δd ) for some δ1 , . . . , δd ∈ J . On the other hand, there exists a unique d-tuple (x1 , . . . , xd ) ∈ J ⊕ d satisfying the equation

  φr (ˆe 1 + x1 , . . . , eˆ d + xd )C = (ˆe 1 + x1 , . . . , eˆ d + xd ). Indeed, the d-tuple t  

 φri (δ1 ), . . . , φri (δd ) φ(C ) · · · φ i −1 (C )φ i (C )

i =0

is stable for sufficiently large t by assumption and this limit gives a unique solution of the equation. Then we have









p (ˆe 1 + x1 ), . . . , p (ˆed + xd ) = φr p (ˆe 1 + x1 ), . . . , p (ˆed + xd ) φ(C ).

Since the d-tuple on the left-hand side is contained in J ⊕ d , we see that this d-tuple is zero and e i → eˆ i + xi defines a section M → E . We can prove the second assertion similarly. 2

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S. Hattori / Journal of Number Theory 129 (2009) 2474–2503 r ,φ

r ,φ

Next we show that the two categories Mod/Σ∞ and Mod/ S ∞ are in fact equivalent. For M ∈ r ,φ Mod/Σ∞ , we associate to it an S-module M by setting M = S ⊗Σ M. We also define its S-submodule

Filr M by





Filr M = Ker M = S ⊗Σ M → S / Filr S ⊗Σ M / Filr M  M / Filr M , where the last isomorphism is induced by the natural isomorphisms of W -algebras





W [u ]/ E (u )r → Σ/ Filr Σ → S / Filr S . r ,φ

These associations induce two functors from Mod/Σ∞ to the category of S-modules, M → M

and M → Filr M. Since the rings S and W [u ]/( E (u )r ) are p-torsion free, we have TorΣ 1 (Σ1 , S ) = r ,φ

r r Σ Σ TorΣ 1 (Σ1 , Σ/ Fil Σ) = 0 and thus Tor1 ( M , S ) = Tor1 ( M , Σ/ Fil Σ) = 0 for any M ∈ Mod/Σ∞ . Hence we see that these two functors are exact. We define φr : Filr M → M as follows. Note that Filr S ⊗Σ M ⊆ M and Filr M is equal to Filr S ⊗Σ M + Im( S ⊗Σ Filr M → M). Set φr : Filr S ⊗Σ M → M to be φr = φr ⊗ φ .

Lemma 3.8. The map φ ⊗ φr : S ⊗Σ Filr M → M induces a φ -semilinear map φr : Im( S ⊗Σ Filr M → M) → M.



Proof. Let z = i si ⊗ mi be in S ⊗Σ Filr M with si ∈ S and mi ∈ Filr M. Let z¯ be its image in M and suppose that z¯ = 0. Write si = si + si with si ∈ Σ and si ∈ Fil p S. Since we have an isomorphism r M S · M  M / Filr Σ · M, we can find elements s( j) ∈ Filr Σ and m( j) ∈ M such that the equality  / Fil m = ( j ) ( j ) holds in M. Then we have s i i i js m

0 = z¯ =



1 ⊗ si mi +

i



si ⊗ mi =



i

s( j ) ⊗ m( j ) +

j



si ⊗ mi

i

in M. On the other hand, the element (φ ⊗ φr )( z) ∈ M is equal to



  1 ⊗ φr s( j ) m( j ) +

   φ si ⊗ φr (mi ).

j

i



Since φ = p r φr , this equals φr (

j

s( j ) ⊗ m( j ) +





i si

⊗ mi ) = 0. 2

Lemma 3.9. The maps φr and φr patch together and define a φ -semilinear map φr : Filr M → M. r r   Proof.  Since φr and φr coincide on Im(Fil S ⊗Σ Fil M → M), it is enough to show  that 1 ⊗ φr (m) = φr ( i si ⊗ mi ) for any m ∈ Filr M, si ∈ Filr S and mi ∈ M satisfying 1 ⊗ m = i si ⊗ mi in M. As  ( j) ( j) for some s( j ) ∈ Filr Σ in the proof of Lemma 3.8, the element m can be written as m = js m

and m( j ) ∈ M. By assumption, we have follows. 2



i si

⊗ mi =



j

s( j ) ⊗ m( j ) in Filr S ⊗Σ M. Hence the lemma

r ,φ

r ,φ

Then we see that this construction defines a functor MΣ∞ : Mod/Σ∞ → Mod/ S ∞ . r ,φ

r ,φ

Lemma 3.10. The functor MΣ∞ induces an equivalence of categories Mod/Σ1 → Mod/ S 1 .

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Proof. Consider the diagram of functors

r ,φ

Mod/Σ1

MΣ∞

r ,φ

Mod/ S 1 T0

T 0,Σ

r ,φ

Mod ˜ . /S1

From the definition, we see that this diagram is commutative. By Lemma 3.1, the downward arrows are equivalences of categories. Thus the lemma follows. 2 Then a dévissage argument as in [15, Proposition 1.1.11] shows the following corollary. r ,φ

r ,φ

Corollary 3.11. The functor MΣ∞ : Mod/Σ∞ → Mod/ S ∞ is fully faithful. To show the essential surjectivity of the functor MΣ∞ , we define another functor M S∞ : r ,φ r ,φ r ,φ r ,φ Mod/S → Mod/Σ∞ which is defined in a similar way to the functor MS∞ : Mod/S → Mod/ S ∞ . ∞

∞

For an S-module M in Mod/S , we associate to it a Σ -module M ∈  Mod/Σ as follows: r ,φ

r ,φ

∞

• M = Σ ⊗φ,S M,

1⊗φ

• Filr M = Ker( M −−−→ Σ ⊗S M → (Σ/ Filr Σ) ⊗S M), 1⊗φ φr ⊗1 • φr : Filr M −−−→ Filr Σ ⊗S M −− −→ Σ ⊗φ,S M = M. r ,φ

r ,φ

We can check that this defines an exact functor Mod/S → Mod/Σ∞ as in the proof of [15, Proposi∞ tion 1.1.11]. We let this functor be denoted by M S∞ . Lemma 3.12. The diagram of functors

M S∞

r ,φ

Mod/S

∞

MS∞

r ,φ

Mod/Σ∞ MΣ∞ r ,φ

Mod/ S ∞

is commutative. r ,φ

Proof. For M ∈ Mod/S , put M = M S∞ (M) and M = MS∞ (M). Then M = S ⊗Σ M as an Sr

∞

module. Let Fil M and φr : Filr M → M denote the filtration and Frobenius structure defined by the

ˆ r M and φˆr : Filˆ r M → M denote those defined by MΣ . functor MS∞ . We also let Fil ∞

ˆ r M. Conversely, let z be an element of Filr M. Note that The S-module Filr M contains Fil

ˆ r M. Thus, to show z ∈ Filˆ r M, we may assume that z ∈ Im( M → M). Then the comFil S · M ⊆ Fil mutative diagram whose right vertical arrow is an isomorphism p

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M = Σ ⊗φ,S M

M = S ⊗φ,S M

1⊗φ

Σ ⊗S M

Σ/ Filr Σ ⊗S M

S ⊗S M

S / Filr S ⊗S M

1⊗φ

ˆ r M and hence Filr M = Filˆ r M. From the definition, we also implies that z ∈ Im(Filr M → Filr M) ⊆ Fil ˆ can show φr = φr . This implies the lemma. 2 r ,φ

r ,φ

Proposition 3.13. The functor MΣ∞ : Mod/Σ∞ → Mod/ S ∞ is an equivalence of categories. Proof. Since the functor MS∞ is an equivalence of categories for p  3 [9, Theorem 2.3.1], Corollary 3.11 and Lemma 3.12 imply the proposition in this case. For r = 0, put κΣ = κ ∩ Σ , where κ = Ker( S → W ). Then, by using a natural isomorphism Σ  W Ju , u ep / p K, we can show that the 0,φ

φ

functor M → M /κΣ M defines an equivalence of categories Mod/Σ∞ → Mod/ W ∞ , as in the case of the 0,φ

category Mod/ S ∞ . Since the diagram 0,φ

Mod/Σ∞

MΣ∞

0,φ

Mod/ S ∞

φ

Mod/ W ∞ is commutative and the downward arrows are equivalences of categories, the proposition follows also for p = 2. 2 r ,φ

r ,φ

Remark 3.14. We can also define a fully faithful functor MΣ : Mod/Σ → Mod/ S in a similar way to MΣ∞ and prove that this is an equivalence of categories. Indeed, the claim for p  3 follows 0,φ

from [9, Theorem 2.2.1]. Let M be in Mod/ S and e 1 , . . . , ed be a basis of M over S. Let C ∈ GLd ( S ) be the matrix such that

φ(e 1 , . . . , ed ) = (e 1 , . . . , ed )C . Then the elements φ(e 1 ), . . . , φ(ed ) also form a basis of M and

    φ φ(e 1 ), . . . , φ(ed ) = φ(e 1 ), . . . , φ(ed ) φ(C ). Since φ( S ) ⊆ Σ , the Σ -module M defined by M = Σφ(e 1 ) ⊕ · · · ⊕ Σφ(ed ) is stable under φ . Hence 0,φ we see that M ∈ Mod/Σ and M = MΣ ( M ). r ,φ

Proposition 3.15. Let M be an object of Mod/Σ∞ and set M = MΣ∞ ( M ). Then there exists a natural isomorphism of G K ∞ -modules

HomΣ,Filr ,φr ( M , A crys,∞ ) → Hom S ,Filr ,φr (M, A crys,∞ ). Moreover, this induces for any n an isomorphism of G K n -modules









HomΣ,Filr ,φr M , W nPD (O˜ K¯ ) → Hom S ,Filr ,φr M, W nPD (O˜ K¯ ) .

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Proof. By definition, M = S ⊗Σ M and we have a natural isomorphism

HomΣ ( M , A crys,∞ ) → Hom S (M, A crys,∞ ). From the definition, we can check that this isomorphism induces the map in the proposition, which is injective. To prove the bijectivity, by dévissage we may assume that pM = 0. Then both sides of this injection have the same cardinality by Lemma 3.4 and the first assertion follows. Since the sequence pn

0 → W nPD (O˜ K¯ ) → A crys,∞ −→ A crys,∞ → 0 r ,φ of the category  Mod/Σ is exact, the first assertion implies the second one.

2

4. A method of Abrashkin In this section, we study the G K n -module HomΣ,Filr ,φr ( M , W nPD (O˜ K¯ )) following Abrashkin [3].

Let p and 0  r < p − 1 be as before. We fix a system of p-power roots of unity {ζ pn }n∈Z0 in K¯ p

such that ζ p = 1 and ζ pn = ζ pn+1 for any n, and set an element elements [ε ] − 1 and [ε

1/ p

] − 1 are topologically nilpotent in W ( R ). The element of W ( R )

 



ε of R to be (ζ pn )n∈Z0 . Then the

t = [ε ] − 1 /











ε 1/ p − 1 = 1 + ε 1/ p + ε 1/ p

2

p −1  + · · · + ε 1/ p

is a generator of the principal ideal Ker(θ). We define an element a ∈ W ( R ) to be

a=

  p −2 k =1

p −1 ((−1) p −1−k p −1 C k − 1)[ε 1/ p ]k

−1

( p  3), ( p = 2),

p −1 C k = ( p − 1)!/(k!( p − 1 − k)!) is the binomial coefficient. Note that the coefficient of [ε 1/ p ]k in each term is an integer. The element a is invertible in the ring W ( R ), since θ(a) = × (ζ p − 1) p −1 / p ∈ OC and the ideal Ker(θ) is topologically nilpotent in W ( R ). The element Z = ([ε ] − 1) p −1 / p of A crys is topologically nilpotent and we have φ(t ) = p ( Z − φ(a)). Consider the formal power series ring W ( R )Ju  K with the (t , u  )-adic topology and the continuous ˆ denote the image of this homoring homomorphism W ( R )Ju  K → A crys which sends u  to Z . Let A ˆ morphism. Then we see that the ring A is (t , Z )-adically complete. Since we have Z = at p −1 + t p / p, ˆ and topologically nilpotent in this subring. the element t p / p of A crys is contained in the subring A ˆ as a Σ -algebra by u → [π ]. Put Fili Aˆ = (t i , Z ) for 0  i  p − 1. The Hence we can consider the ring A ˆ and satisfies φ(Fili Aˆ ) ⊆ p i Aˆ for 0  i  p − 1. Set Frobenius endomorphism φ of A crys preserves A r ,φ φr = p −r φ| r ˆ . Then we can consider the ring Aˆ also as an object of the category  Mod . Put Aˆ n =

where

/Σ

Fil A

ˆ / pn Aˆ and Aˆ ∞ = Aˆ ⊗W K 0 / W . We include here a proof of the following lemma stated in [3, SubsecA tion 3.2].

ˆ induces isomorphisms of W ( R )-algebras W ( R )/ Lemma 4.1. The natural inclusion W ( R ) → A (([ε ] − 1) p −1 ) → Aˆ /( Z ) and W n ( R )/(([ε ] − 1) p −1 ) → Aˆ n /( Z ). Proof. For a subring B of A crys , put





I [s] B = x ∈ B φ i (x) ∈ Fils A crys for any i



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as in [12, Subsection 5.3]. Then we have I [s] W ( R ) = ([ε ] − 1)s W ( R ) and the natural ring homomorphism

W ( R )/ I [s] W ( R ) → A crys / I [s] A crys is an injection [12, Propositions 5.1.3, 5.3.5]. Since the element Z is contained in the ideal I [ p −1] A crys , this injection factors as

ˆ /( Z ) → A crys / I [ p −1] A crys . W ( R )/ I [ p −1] W ( R ) → A Hence the former arrow is an isomorphism and the lemma follows.

2

ˆ / Filr Aˆ is p-torsion free and pn Filr Aˆ = Filr Aˆ ∩ pn A. ˆ Thus we can also consider Aˆ n Therefore A r ,φ  ˆ and A ∞ as objects of the category Mod/Σ . The absolute Galois group G K ∞ acts naturally on these Σ -modules. Lemma 4.2. We have a natural decomposition as an R-module

 

ˆ 1 = R / t p ⊕ ( Z ). A ˆ We claim that this induces an injection R /(t p ) → Aˆ 1 . Proof. Consider the natural inclusion W( R ) → A. ˆ then its image in A crys / p A crys is Let x be in the ring R. If the element [x] ∈ W ( R ) is contained in p A, zero. We have an isomorphism of R-algebras



p

p



R [Y 1 , Y 2 , . . .]/ t p , Y 1 , Y 2 , . . . → A crys / p A crys i

which sends Y i to the image of t p / p i !. Thus the element x is contained in the ideal (t p ). Conversely, if v R (x)  p, then we have

  p −1 [x] = w [ε ] − 1 + p w ˆ Now we have the commutative diagram of for some w , w  ∈ W ( R ) and this implies [x] ∈ p A. R-algebras

ˆ1 A

R /(t p ) f

ˆ 1 /( Z ) A ˆ 1 /( Z ) is an isomorphism by Lemma 4.1. Hence the lemma follows. and the map f : R /(t p ) → A

2

Since r < p − 1, from this lemma we can show the following lemma as in the proof of [6, Lemme 2.3.1.3]. Lemma 4.3. The functor

ˆ ∞) M → HomΣ,Filr ,φr ( M , A r ,φ

from Mod/Σ∞ to the category of G K ∞ -modules is exact.

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r ,φ

Corollary 4.4. For any M ∈ Mod/Σ∞ , the natural map

ˆ ∞ ) → HomΣ,Filr ,φ ( M , A crys,∞ ) HomΣ,Filr ,φr ( M , A r is an isomorphism of G K ∞ -modules. Moreover, for any n, we have an isomorphism of G K ∞ -modules





ˆ n ) → HomΣ,Filr ,φ M , A crys / pn A crys . HomΣ,Filr ,φr ( M , A r Proof. Let us prove the first assertion. By Lemmas 3.3 and 4.3, we may assume pM = 0. Consider the commutative diagram of rings

ˆ1 A

A crys / p A crys

R /(t p −1 )

ˆ 1 and A crys / p A crys , respectively. whose downward arrows are defined by modulo Fil p −1 of the rings A

ˆ 1 ) = 0 and similarly for the ring A crys / p A crys . Thus these two Since r < p − 1, we have φr (Fil p −1 A surjections induce on the ring R /(t p −1 ) the same structure of a filtered φr -module over Σ . By Corollary 3.7, we have a commutative diagram

ˆ 1) HomΣ,Filr ,φr ( M , A

HomΣ,Filr ,φr ( M , A crys / p A crys )

HomΣ,Filr ,φr ( M , R /(t p −1 )) whose downward arrows are isomorphisms. This concludes the proof of the first assertion. Since we have an exact sequence n

p ˆ n → Aˆ ∞ − 0→ A → Aˆ ∞ → 0 r ,φ in the category  Mod/Σ , the second assertion follows.

2

ˆ n satisfies the condition of Corollary 3.7, the Σ -algebra Aˆ n /( Z ) is naturally Since the ideal ( Z ) of A r ,φ considered as an object of  Mod/Σ . We also give the ring W n ( R )/(([ε ] − 1) p −1 ) the structures of a

Σ -algebra and a filtered φr -module over Σ induced from those of Aˆ n /( Z ) by the isomorphism in Lemma 4.1. The map

 p −1   Σ → W n ( R )/ [ε ] − 1 sends the element u ∈ Σ to the image of [π ] in the ring on the right-hand side. Put v = t / E ([π ]) ∈ W ( R )× . As for the element Y ∈ Σ , the equality



 p −1

Y = −av −1 E [π ]

+ v −p Z

ˆ Hence the above homomorphism sends the element Y to the image of −av −1 E ([π ]) p −1 . holds in A.

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Consider the surjective ring homomorphism

R → O˜ K¯ , x = (x0 , x1 , . . .) → xn and the induced surjection βn : W n ( R ) → W n (O˜ K¯ ). Let





J = (x0 , . . . , xn−1 ) ∈ W n ( R ) v R (xi )  pn for any i



be the kernel of the latter surjection. Lemma 4.5. The ideal J is contained in the ideal (([ε ] − 1) p −1 ) of the ring W n ( R ). Proof. Write the element ([ε ] − 1) p −1 also as x = (x0 , . . . , xn−1 ) ∈ W n ( R ) with v R (x0 ) = p. Take an element z = ( z0 , . . . , zn−1 ) of the ideal J . We construct y ∈ W n ( R ) such that xy = z. By induction, it is enough to show that if z0 = · · · = zi −1 = 0 for some 0  i  n − 1 and (x0 , . . . , xi )(0, . . . , 0, y i ) = (0, . . . , 0, zi ) in W i +1 ( R ), then x(0, . . . , 0, y i , 0, . . . , 0) ∈ J . Let us write this element as (0, . . . , 0, w i , . . . , w n−1 ) with w i = zi . We have v R ( y i )  pn − p i +1 . In the ring of Witt vectors W n (F p [ X 0 , . . . , X n−1 , Y 0 , . . . , Y n−1 ]), the k-th entry of the vector

( X 0 , . . . , Xn−1 )(0, . . . , 0, Y i , 0, . . . , 0) pi

p k −i

is X k−i Y i

for any k  i. Thus we have v R ( w k )  pn .

2

˜ ¯ ). By the above lemma, Note that the elements [ζ pn ] − 1 and [ζ pn+1 ] − 1 are nilpotent in W n (O K we have an isomorphism of rings



W n ( R )/ [ε ] − 1

 p −1 

 p −1   → W n (O˜ K¯ )/ [ζ pn ] − 1 .

¯ n, p −1 denote the ring on the right-hand side and give the ring A¯ n, p −1 the structure of We let A a filtered φr -module over Σ induced by this isomorphism. For an algebraic extension F of K , we put

  b F = x ∈ O F v K (x) > er /( p − 1) . Note that the ring O F /b F is killed by p. We consider the ring of Witt vectors W n (O F /b F ) as a W n (O F )-algebra by the natural ring surjection W n (O F ) → W n (O F /b F ) and as a W n -algebra by twisting the natural action by σ −n , as before. For a ring B and its ideal I , we define an ideal W n ( I ) of the ring W n ( B ) to be







W n ( I ) = (x0 , . . . , xn−1 ) ∈ W n ( B ) xi ∈ I for any i . Put F n = K n (ζ pn+1 ). For an algebraic extension F of F n in K¯ , the elements [ζ pn ] − 1 and [ζ pn+1 ] − 1 of W n (m F ) are topologically nilpotent non-zero divisors in W n (O F ). Let the ring



r

W n (O F /b F )/ [ζ pn ] − 1 W n (m F /b F )

¯ n, F ,r + . We also put A¯ n,r + = A¯ ¯ . be denoted by A n, K ,r +

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Lemma 4.6. The ideal ([ζ pn ] − 1)r W n (m F ) of W n (O F ) contains the ideal W n (b F ) for any r ∈ {0, . . . , p − 2}. We also have (([ζ pn ] − 1) p −1 ) ⊇ W n ( p O F ). Proof. The proof is similar to the proof of Lemma 4.5. Let us show the first assertion. Since this is trivial for r = 0, we may assume r  1. Put x = (x0 , . . . , xn−1 ) = ([ζ pn ] − 1)r ∈ W n (O F ). Then we have v p (x0 ) = r /( pn−1 ( p − 1)). By induction, it is enough to show that for 0  i  n − 1, if (x0 , . . . , xi )(0, . . . , 0, y i ) ∈ W i +1 (b F ), then y i ∈ m F and x(0, . . . , 0, y i , 0, . . . , 0) ∈ W n (b F ). By assumption, we have

v p ( yi ) >

r



p−1

1−

1 p n − i −1

  0.

Put (0, . . . , 0, w i , . . . , w n−1 ) = x(0, . . . , 0, y i , 0, . . . , 0). We show w l ∈ b F for any l by induction. Indeed, let us suppose that w l ∈ b F for any i  l  k − 1 with some i + 1  k  n − 1. We have the equality p k −i  p k x0

pi yi

p k −1

+ px1

   p k −i p k − i −1 + · · · + pk xk = p i w i + p i +1 w i +1 + · · · + pk w k . pk−l

Since r  1, we have ( pk−l − 1)r /( p − 1)  k − l for 0  l  k − 1. This implies v p ( pl w l ) > k + r /( p − 1) for 0  l  k − 1. The valuation of the left-hand side of the above equality also satisfies this inequality. Thus we have v p ( w k ) > r /( p − 1) and the assertion follows. We can show the second assertion similarly. 2 By this lemma, the natural surjections of rings









r r ¯ n, F ,r + W n (O F )/ [ζ pn ] − 1 W n (m F ) → W n (O F / p O F )/ [ζ pn ] − 1 W n (m F / p O F ) → A

are isomorphisms. Then we see that the natural injection F → K¯ induces an injection of rings ¯ n, F ,r + → A¯ n,r + . A Write Z n for the image of the element Z of A crys in W nPD (O˜ K¯ ). Then we have a commutative diagram of Σ -algebras

ˆn A

A crys / pn A crys 

W nPD (O˜ K¯ )

W n ( R )/(([ε ] − 1) p −1 )

∼

ˆ n /( Z ) A



¯ n , p −1 A

W nPD (O˜ K¯ )/( Z n ),

¯ n,r + A where all the vertical arrows are surjections satisfying the condition of Corollary 3.7. Hence this is r ,φ also a commutative diagram in  Mod/Σ . Note that these rings and homomorphisms are independent

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¯ n,r + = E ([πn ])r A¯ n,r + and φr ( E ([πn ])r y ) = of the choice of a system {ζ pn }n∈Z0 . We also note that Filr A r ¯ n,r + induced from that ¯ c φ( y ) for any y ∈ A n,r + , where φ denotes the Frobenius endomorphism of A r ,φ

of the ring W n (O K¯ /b K¯ ). Moreover, let M be an object of Mod/Σ∞ . Then, by Corollaries 3.7 and 4.4, we have a natural isomorphism of abelian groups





¯ n,r + ). HomΣ,Filr ,φr M , W nPD (O˜ K¯ ) → HomΣ,Filr ,φr ( M , A Next we investigate the module on the right-hand side of this isomorphism, and prove this is in fact an isomorphism of G F n -modules. Consider the element E ([πn ]) ∈ W n (O F n / p O F n ) and let us fix its lift γˆ ∈ W n (O F n ) by the natural surjection W n (O F n ) → W n (O F n / p O F n ). Let a ∈ W ( R )× and v = t / E ([π ]) ∈ W ( R )× as before. We let an , tn and v n denote the images of a, t and v by the surjection W ( R ) → W n (O˜ K¯ ) induced by βn , respectively. The elements an and tn of the ring W n (O˜ K¯ ) are contained in the subring W n (O F n / p O F n ). We abusively let them also denote their images by the ¯ n,r + . natural surjections W n (O˜ K¯ ) → W n (O K¯ /b K¯ ) → A Lemma 4.7. The element

tˆn = 1 + [ζ pn+1 ] + [ζ pn+1 ]2 + · · · + [ζ pn+1 ] p −1 =

[ζ pn ] − 1 [ζ pn+1 ] − 1

is divisible by γˆ in the ring W n (O F n ). In particular, γˆ is a non-zero divisor of the ring W n (O K¯ ). Proof. It is enough to show the divisibility in the ring W n (O K¯ ). Note that the element tn is also the image of tˆn by the natural map W n (O F n ) → W n (O F n / p O F n ). Let vˆ n be a lift of v n by the natural ˜ ¯ ). Then we have tˆn − γˆ vˆ n ∈ W n ( p O ¯ ). By Lemma 4.6, there exists surjection W n (O K¯ ) → W n (O K K yˆ ∈ W n (m K¯ ) such that tˆn − γˆ vˆ n = tˆn yˆ . Hence we have tˆn (1 − yˆ ) = γˆ vˆ n . Since yˆ is topologically nilpotent in the ring W n (O K¯ ), the element 1 − yˆ is invertible and the lemma follows. 2

¯ n,r + (resp. A¯ n, p −1 ) is contained in its subring A¯ n, F n ,r + (resp. Lemma 4.8. The image of Y ∈ Σ in the ring A W n (O F n / p O F n )/(([ζ pn ] − 1) p −1 )). Proof. We have the equality





E [πn ] v n = tn = 1 + [ζ pn+1 ] + [ζ pn+1 ]2 + · · · + [ζ pn+1 ] p −1 in the ring W n (O˜ K¯ ). Note that any element v n ∈ W n (O˜ K¯ ) satisfying the same equality is invertible and thus the elements ( v n )−1 E ([πn ]) are equal to each other. Since Y = −an v n−1 E ([πn ]) p −1 in the ¯ n,r + and A¯ n, p −1 , it suffices to construct an element v n of the ring W n (O F n / p O F n ) such that rings A the equality E ([πn ]) v n = tn holds. This follows from Lemma 4.7. 2

¯ n, p −1 and A¯ n,r + are compatiFrom this lemma, we see that the natural G F n -actions on the rings A ble with the filtered φr -module structures over Σ . In the big commutative diagram above, the lowest horizontal arrow and lower right vertical arrow are G K -linear by definition. Hence we have shown the following proposition. r ,φ

Proposition 4.9. Let M be an object of Mod/Σ∞ . Then the map





¯ n,r + ) HomΣ,Filr ,φr M , W nPD (O˜ K¯ ) → HomΣ,Filr ,φr ( M , A is an isomorphism of G F n -modules.

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Let M be as in the proposition. Let e 1 , . . . , ed be a system of generators of M as in Lemma 3.5 and C = (c i , j ) ∈ M d (Σ) be a matrix representing φr as in Corollary 3.6. Consider the surjection Σ ⊕ d → M defined by (s1 , . . . , sd ) → s1 e 1 + · · · + sd ed and let (s1,1 , . . . , s1,d ), . . . , (sq,1 , . . . , sq,d ) be a system of generators of its kernel. Then the underlying G F n -set of the G F n -module

¯ n,r + ) HomΣ,Filr ,φr ( M , A ¯ n,r + such that the following three conditions is identified with the set of d-tuples (¯x1 , . . . , x¯ d ) in A hold: • sl,1 x¯ 1 + · · · + sl,d x¯ d = 0 for any l, • c 1,i x¯ 1 + · · · + cd,i x¯ d ∈ Filr A¯ n,r + for any i, • the following equality holds:

⎧ ⎨ φr (c 1,1 x¯ 1 + · · · + cd,1 x¯ d ) = x¯ 1 , .. ⎩. φr (c 1,d x¯ 1 + · · · + cd,d x¯ d ) = x¯ d . ¯ n,r + by the natural We choose lifts cˆ , cˆ i , j and sˆ i , j in W n (O F n ) of the images of c, c i , j and si , j in A ring homomorphism

¯ n,r + , W n (O K¯ ) → W n (O˜ K¯ ) → W n (O K¯ /b K¯ ) → A respectively. Recall that we have already chosen a lift γˆ ∈ W n (O F n ) of E ([πn ]) ∈ W n (O F n / p O F n ). p Fix a polynomial Φi ∈ Z[ X 0 , . . . , X n−1 ] such that Φi ≡ X i mod p. This induces for any commutative ring B a map Φ = (Φ0 , . . . , Φn−1 ) : W n ( B ) → W n ( B ) which is a lift of the Frobenius endomorphism on W n ( B / p B ). In particular, set B to be the polynomial ring Z[ X 0 , . . . , X n−1 , Y 0 , . . . , Y n−1 ]. Put X = ( X 0 , . . . , Xn−1 ) and Y = (Y 0 , . . . , Y n−1 ) in the ring W n ( B ). Then we see that there exist elements U 0 , . . . , U n−1 and U 0 , . . . , U n −1 of the polynomial ring B such that

Φ( X + Y ) = Φ( X ) + Φ(Y ) + ( pU 0 , . . . , pU n−1 ),   Φ( X Y ) = Φ( X )Φ(Y ) + pU 0 , . . . , pU n −1 in the ring W n ( B ).

¯ n,r + satisfying the above three conditions uniquely lifts to a Proposition 4.10. Every d-tuple (¯x1 , . . . , x¯ d ) in A d-tuple (ˆx1 , . . . , xˆ d ) in W n (O K¯ ) such that • sˆl,1 xˆ 1 + · · · + sˆl,d xˆ d ∈ ([ζ pn ] − 1)r W n (m K¯ ) for any l, • cˆ 1,i xˆ 1 + · · · + cˆ d,i xˆ d ∈ γˆ r W n (O K¯ ) for any i, • the following equality holds:

 ⎧ r  r ⎨ c.ˆ Φ (ˆc 1,1 xˆ 1 + · · · + cˆd,1 xˆ d )/γˆ = xˆ 1 , ..  ⎩ r  cˆ Φ (ˆc 1,d xˆ 1 + · · · + cˆ d,d xˆ d )/γˆ r = xˆ d . ¯ n,r + is Proof. Fix a lift xˆ i of x¯ i in W n (O K¯ ). Recall that the kernel of the surjection W n (O K¯ ) → A equal to the ideal ([ζ pn ] − 1)r W n (m K¯ ). The first condition in the proposition holds automatically for (ˆx1 , . . . , xˆ d ). By Lemma 4.7, the element cˆ 1,i xˆ 1 + · · · + cˆ d,i xˆ d is contained in γˆ r W n (O K¯ ) for any i. Since

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¯ n,r + → A¯ n,r + satisfies φr ( E ([πn ])r x¯ ) = cr φ(¯x) for any x¯ ∈ A¯ n,r + , we have the map φr : Filr A

⎧  r   r r ⎪ ⎨ c.ˆ Φ (ˆc 1,1 xˆ 1 + · · · + cˆd,1 xˆ d )/γˆ = xˆ 1 + [ζ pn ] − 1 δˆ1 , .. ⎪ ⎩ cˆ r Φ (ˆc xˆ + · · · + cˆ xˆ )/γˆ r  = xˆ + [ζ n ] − 1r δˆ p 1,d 1 d,d d d d for some δˆ1 , . . . , δˆd ∈ W n (m K¯ ). It suffices to show that there exists a unique d-tuple ( yˆ 1 , . . . , yˆ d ) in W n (m K¯ ) such that





r







r







r



cˆ r Φ cˆ 1,i xˆ 1 + [ζ pn ] − 1 yˆ 1 + · · · + cˆ d,i xˆ d + [ζ pn ] − 1 yˆ d /γˆ r = xˆ i + [ζ pn ] − 1 yˆ i for any i. For this, we need the following lemma. Lemma 4.11. Let N be a complete discrete valuation field and m N be the maximal ideal of N. Let 1 , . . . , d be in m N . Let P 1 , . . . , P d and P 1 , . . . , P d be elements of O N JY 1 , . . . , Y d K such that P i ∈ (Y 1 , . . . , Y d )2 . Then the equation

⎧  ⎨ .Y 1 − P 1 (Y 1 , . . . , Y d ) − 1 P 1 (Y 1 , . . . , Y d ) = 0, .. ⎩ Y d − P d (Y 1 , . . . , Y d ) − d P d (Y 1 , . . . , Y d ) = 0 has a unique solution in m N . Proof. By assumption, we see that for any integer l  1, a d-tuple ( y 1 , . . . , yd ) in m N /mlN satisfying

the above equation lifts uniquely to a d-tuple in m N /mlN+1 satisfying the same equation. Thus the lemma follows. 2

¯ n,r + is equal to Let us write as yˆ i = ( yˆ i ,0 , . . . , yˆ i ,n−1 ). Since the image of Φ(([ζ pn+1 ] − 1)r ) in A

([ζ pn ] − 1)r , we can find bˆ ∈ W n (O K¯ ) such that

r   r  Φ [ζ pn ] − 1 /γˆ r = [ζ pn ] − 1 bˆ . Then there exist polynomials U i ,m over O K¯ of the indeterminates Y = (Y i ,m )1i d, 0mn−1 such that the equation we have to solve is

















r r r xˆ i + [ζ pn ] − 1 yˆ i = xˆ i + [ζ pn ] − 1 δˆi + [ζ pn ] − 1 bˆ cˆ r Φ(ˆc 1,i )Φ( yˆ 1 ) + · · · + Φ(ˆcd,i )Φ( yˆ d )

  + pU i ,0 ( yˆ ), . . . , pU i ,n−1 ( yˆ )

for any i, where we put yˆ = ( yˆ i ,m )1i d, 0mn−1 . As in the proof of Lemma 4.6, we see that, for any elements P 0 , . . . , P n−1 of the polynomial ring O K¯ [Y ], we can uniquely find elements Q 0 , . . . , Q n−1 of this ring such that the coefficients of these polynomials are in the maximal ideal m K¯ and the equality

r  ( p P 0 , . . . , p P n−1 ) = [ζ pn ] − 1 ( Q 0 , . . . , Q n−1 ) holds in the ring of Witt vectors W n (O K¯ [Y ]). Therefore, this equation is equivalent to the equation









yˆ i = δˆi + bˆ cˆ r Φ(ˆc 1,i )Φ( yˆ 1 ) + · · · + Φ(ˆcd,i )Φ( yˆ d ) + V i ,0 ( yˆ ), . . . , V i ,n−1 ( yˆ ) ,

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where V i ,m is a polynomial of Y over O K¯ whose coefficients are in the maximal ideal m K¯ . From the definition of Φ , we see that yˆ = ( yˆ i ,m )i ,m is a solution of a system of equations

Y i ,m − P i ,m (Y ) − i ,m P i,m (Y ) = 0 satisfying the condition of Lemma 4.11 for a sufficiently large finite extension N of K . Then, by this lemma, we can solve the equation uniquely in m K¯ . 2

¯ n, F ,r + . By Lemma 4.8, we can Let F be an algebraic extension of F n in K¯ and consider the ring A

¯ n,r + . Put Filr A¯ n, F ,r + = E ([πn ])r A¯ n, F ,r + . Then Lemma 4.7 consider this ring as a Σ -subalgebra of A implies that

¯ n, F ,r + ∩ Filr A¯ n,r + = Filr A¯ n, F ,r + . A ¯ n,r + preserves the subalgebra A¯ n, F ,r + and Moreover, the Frobenius endomorphism φ of the ring A r ¯ ¯ n, F ,r + → A¯ n, F ,r + . Hence A¯ n, F ,r + is ¯ thus φr : Fil A n,r + → A n,r + induces a φ -semilinear map φr : Filr A ¯ n,r + in the category  Modr ,φ . For M ∈ Modr ,φ , let us set a subobject of A /Σ /Σ∞ ∗ ¯ T crys ,πn , F ( M ) = HomΣ,Filr ,φr ( M , A n, F ,r + ).

We see that

¯ n,r + = A¯ ¯ A n, K ,r + =



¯ n, F ,r + A

F /Fn r ,φ in  Mod/Σ and thus we have a natural identification of abelian groups

∗ T crys (M ) = ,πn , K¯



∗ T crys ,πn , F ( M ).

F /Fn

The absolute Galois group G F n acts on the abelian group on the left-hand side. Lemma 4.12. Let F be an algebraic extension of F n in K¯ . Then the G F -fixed part T ∗

crys,πn , K¯

∗ to T crys ,πn , F ( M ).

Proof. From Proposition 4.10, we see that the elements of T ∗

crys,πn , K¯

( M )G F is equal

( M ) correspond bijectively to the

d-tuples in W n (O K¯ ) satisfying the three conditions in this proposition. The uniqueness assertion of the proposition shows that g ∈ G F fixes such a d-tuple in W n (O K¯ ) if and only if g fixes its image ¯ n,r + . Hence an element of T ∗ in A ¯ ( M ) is fixed by G F if and only if it is contained in the image crys,πn , K

of W n (O F ). Thus the lemma follows.

2

Corollary 4.13. Let L n be the finite Galois extension of F n corresponding to the kernel of the map





∗ G F n → Aut T crys (M ) . ,πn , K¯

Then an algebraic extension F of F n in K¯ contains L n if and only if ∗ ∗ #T crys ,πn , F ( M ) = #T crys,πn , K¯ ( M ).

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Proof. An algebraic extension F of F n contains L n if and only if the action of G F on T ∗ ∗ ∗ trivial. By Lemma 4.12, this is equivalent to T crys ,πn , F ( M ) = T crys,πn , K¯ ( M ).

crys,πn , K¯

2

( M ) is

5. Ramification bound r ,φ, N

In this section, we prove Theorem 1.1. Let M be an object of Mod/ S ∞ which is killed by pn and let L be the finite Galois extension of K corresponding to the kernel of the map





∗ G K → Aut T st ,π (M) .

Then the theorem is equivalent to the inequality u L / K  u ( K , r , n), where u L / K denotes the greatest ∗ (M) is upper ramification break of the Galois extension L / K [10]. For r = 0, the G K -module T st ,π unramified and the assertion is trivial. Thus we may assume p  3 and r  1. Let L n be the finite Galois extension of F n corresponding to the kernel of the map





∗ G F n → Aut T st ,π (M) . r ,φ

Since F n is Galois over K , the extension L n = L F n is also a Galois extension of K . Let M ∈ Mod/Σ∞ be the filtered φr -module over Σ which corresponds to M by the equivalence MΣ∞ of Proposition 3.13. Then Propositions 3.15 and 4.9 show that L n is also the finite extension of F n cut out by the G F n -module T ∗ ¯ ( M ). It is enough to prove the inequality u Ln / K  u ( K , r , n). crys,πn , K

Before proving this, we state some general lemmas to calculate the ramification bound. Let N be a complete discrete valuation field of positive residue characteristic, v N be its valuation normalized as v N ( N × ) = Z and N sep be its separable closure. We extend v N to any algebraic closure of N. Lemma 5.1. Let f ( T ) ∈ O N [ T ] be a separable monic polynomial and z1 , . . . , zd be the zeros of f in O N sep . Suppose that the set { v N ( zk − zi ) | k = 1, . . . , d, k = i } is independent of i. Put



sf =

v N ( zk − zi ) and

k=1,...,d k=i

α f = sup v N (zk − zi ), k=1,...,d k=i

which are independent of i by assumption. If j > s f + α f , then we have the decomposition











x ∈ O N sep v N f (x)  j =

 





x ∈ O N sep v N (x − zi )  j − s f .

i =1,...,d

Otherwise, the set on the left-hand side contains







x ∈ O N sep v N (x − zi )  α f ,

which contains at least two zeros of f . Proof. A verbatim argument in the proof of [1, Lemma 6.6] shows the claim.

2

Corollary 5.2. Let f ( T ) be as above and put B = O N [ T ]/( f ( T )). Let us write the N-algebra N  = B ⊗ON N as the product N 1 × · · · × N t of finite separable extensions N 1 , . . . , N t of N. If j > s f + α f , then the j-th upper ( j)

numbering ramification group [1], which we let be denoted by G N , is contained in G N i for any i. Moreover, if N  is a field and B coincides with

( j)

O N  , then j > s f + α f if and only if G N ⊆ G N  .

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Proof. Note that the algebra B is finite flat and of relative complete intersection over O N . By the previous lemma, the conductor c ( B ) of the O N -algebra B [1, Proposition 6.4] is equal to s f + α f . Thus we have the inequality

c (O N 1 × · · · × O Nt )  c ( B ) = s f + α f by the definition of the conductor and a functoriality of the functor F j defined in [1]. This implies the corollary. 2 Corollary 5.3. We have the inequality

u K (ζ

p n +1

)/ K

1−



1 e ( K (ζ p )/ K )



1

+e n+

p−1

,

where e ( K (ζ p )/ K ) denotes the relative ramification index of K (ζ p ) over K . Proof. Since the Herbrand function is transitive and the finite extension K (ζ p ) is tamely ramified over K , it is enough to show the inequality

u K (ζ

    e K (ζ ) n+ p )/ K (ζ p ) p n +1



1

.

p−1

n

Put N = K (ζ p ) and f ( T ) = T p − ζ p . These satisfy the assumptions of Corollary 5.2. We have s f = ne ( K (ζ p )) and α f = e ( K (ζ p ))/( p − 1) in this case. Hence the corollary follows. 2 Corollary 5.4. Consider the finite Galois extension F n = K n (ζ pn+1 ) of K . Then we have the equality

 u Fn /K = 1 + e n +

1 p−1

 . n

Proof. Applying Corollary 5.2 to the Eisenstein polynomial f ( T ) = T p − π and N = K shows that j > ( j) 1 + e (n + 1/( p − 1)) if and only if G K ⊆ G K n . From Corollary 5.3, we see that if j > 1 + e (n + 1/( p − 1)), ( j)

then G K ⊆ G K (ζ ( j)

p n +1

only if G K ⊆ G F n .

) . Since G F n

= G K n ∩ G K (ζ pn+1 ) , we conclude that j > 1 + e (n + 1/( p − 1)) if and

2

Remark 5.5. Note that this argument also shows the equality

 u K n (ζ pn )/ K = 1 + e n +

1



p−1

.

Next we assume that the residue field of N is perfect. For an algebraic extension F of N, we put

  j a F / N = x ∈ O F v N (x)  j . Let Q be a finite Galois extension of N and consider the property

(P j)

⎧ ⎨ for any algebraic extension F of N, if there exists j an O N -algebra homomorphism O Q → O F /a F / N , ⎩ then there exists an N-algebra injection Q → F

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for j ∈ R0 , as in [10, Proposition 1.5]. Then we have the following proposition, which is due to Yoshida. Here we reproduce his proof for the convenience of the reader. Proposition 5.6. (See [19].)







u Q / N = inf j ∈ R0 the property ( P j ) holds . Proof. By [10, Proposition 1.5(i)], it is enough to show that the property ( P j ) does not hold for j = u Q / N − (e  )−1 with an arbitrarily large e  > 0. As in the proof of [10, Proposition 1.5(ii)], we may assume that Q is totally and wildly ramified over N. Take an arbitrarily large integer e  > 0 with (e  , pe ( Q / N )) = 1. We may also assume that N contains a primitive e  -th root of unity. Set 1/e 

N  = N (π N ) and Q  = Q N  . Note that we have u Q  / N = u Q / N by assumption. From this proposition in [10], we see that for some algebraic extension F of N, there exists an O N -algebra homomorphism j O Q  → O F /a F / N for j = u Q / N − e ( Q  / N )−1 but no N-algebra injection Q  → F . Since Q / N is wildly

ramified, we see that e ( Q / N )u Q / N − 1 > e ( Q / N ). Hence we have u Q / N − e ( Q  / N )−1 > 1  u N  / N and there exists an N-algebra injection N  → F also by this proposition. Thus there exists no N-algebra injection Q → F and the property ( P j ) for Q / N does not hold. Since e ( Q  / N ) = e  e ( Q / N ), the proposition follows. 2 We see from Proposition 5.6 that to bound the greatest upper ramification break u Ln / K , it is enough to show the following proposition.

Proposition 5.7. Let F be an algebraic extension of K . If j > u ( K , r , n) and there exists an O K -algebra homomorphism

η : OLn → O F /a Fj / K , then there exists a K -algebra injection L n → F . Proof. We may assume that F is contained in K¯ . By assumption, we have j > er /( p − 1) and we j

see that the ideal b F = {x ∈ O F | v K (x) > er /( p − 1)} contains a F / K . Thus homomorphism

η induces an O K -algebra

O Ln → O F /b F . j

Since η also induces an O K -algebra homomorphism O F n → O F /a F / K and r  1, from Corollary 5.4 and [10, Proposition 1.5] we get a K -linear injection F n → F . Thus we see that F contains πn and ζ pn+1 . More precisely, we have the following two lemmas. Lemma 5.8. There exists i ∈ Z such that η(πn ) ≡ πn ζ pi n mod b F . Proof. Since the map η is O K -linear, the equality of η(πn ) in O F . Then we have



v K xˆ

pn



−π =

p n −1

 i =0

n

η(πn ) p = π holds in O F /a Fj / K . Set xˆ to be a lift





v K xˆ − πn ζ pi n  j .

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n

Let us apply Lemma 5.1 to f ( T ) = T p − π ∈ O K [ T ]. Then, with the notation of the lemma, we have

sf = 1−

1 pn

+ ne and α f =

1

+

pn

e p−1

.

Since j − s f > er /( p − 1) by assumption, we have

xˆ ≡ πn ζ pi n mod b F for some i.

2

Lemma 5.9. There exists g  ∈ G K such that η(ζ pn+1 ) ≡ g  (ζ pn+1 ) mod b F . Proof. Set N to be the maximal unramified subextension of K (ζ pn+1 )/ K . Since the map O K → O N j

is etale, there exists a K -algebra injection g 0 : N → F such that η(x) ≡ g 0 (x) mod a F / K for any x ∈ O N . Let  be a uniformizer of K (ζ pn+1 ) and f ( T ) ∈ O N [ T ] be the Eisenstein polynomial of  over O N . We let f g0 ( T ) ∈ O N [ T ] denote the conjugate of f by g 0 . Then f g0 satisfies the conditions of Lemma 5.1. By definition we have s f g0 = s f and α f g0 = α f . Since the roots of f g0 ( T ) are conjugates of  over K , Lemma 5.1 implies as in the previous lemma that there exists g  ∈ G K such that g  | N = g 0 and

j −s

η( ) ≡ g  ( ) mod a F / K f . Since O K (ζ pn+1 ) is generated by  over ON , we see that j −s

η(ζ pn+1 ) ≡ g  (ζ pn+1 ) mod a F / K f .

Thus it is enough to check the inequality j − s f > er /( p − 1). Note that s f is equal to the valuation v K (D K (ζ n+1 )/ N ) of the different of the totally ramified Galois extension K (ζ pn+1 )/ N. To bound this, p

put G = Gal( K (ζ pn+1 )/ N (ζ p )) and e  = e ( N (ζ p )/ N ). We have

vK for any







τ ( ) −   v K τ (ζ pn+1 ) − ζ pn+1



τ ∈ G and thus v K (D K (ζ

)/ N (ζ p ) p n +1

)



vK





τ (ζ pn+1 ) − ζ pn+1  ne.

τ =1∈G

We also have the equality v K (D N (ζ p )/ N ) = 1 − 1/e  and hence we get

s f = v K (D K (ζ

p n +1

)/ N )  1 − 1/e

Since e   p − 1, the inequality j − s f > er /( p − 1) holds.



+ ne .

2

Corollary 5.10. There exists g ∈ G K such that η(πn ) ≡ g (πn ) mod b F and η(ζ pn+1 ) ≡ g (ζ pn+1 ) mod b F . Proof. Let i ∈ Z and g  ∈ G K be as in Lemma 5.8 and Lemma 5.9, respectively. Since K n ∩ K (ζ pn+1 ) = K (see for example [17, Lemma 5.1.2]), we can find an element g ∈ G K such that g (πn ) = πn ζ pi n and g (ζ pn+1 ) = g  (ζ pn+1 ).

2

(m)

Lemma 5.11. For m ∈ Z0 , set an ideal b Ln of O Ln to be

 (m) b Ln = x ∈ O Ln v K (x) >

er p m ( p − 1)



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S. Hattori / Journal of Number Theory 129 (2009) 2474–2503

and similarly for F . Then the O K -algebra homomorphism η induces an O K -algebra injection

η(m) : OLn /b(Lmn ) → O F /b(Fm) for any m. Proof. We may assume that L n is totally ramified over K . We write the Eisenstein polynomial of a uniformizer π Ln of L n over O K as 



P ( T ) = T e + c 1 T e −1 + · · · + c e  −1 T + c e  , j where e  = e ( L n / K ). Then z = η(π Ln ) satisfies P ( z) = 0 in O F /a F / K . Let zˆ be a lift of z in O F . Since

j > 1, we have v K (ˆz) = 1/e  . The condition i > e ( L n )r /( pm ( p − 1)) is equivalent to the condition

 

v K zˆ i > Thus the claim follows.

e ( L n )r p m ( p − 1)

·

1 e

=

er p m ( p − 1)

.

2

Since L n contains F n , we can consider the ring



r

¯ n, Ln ,r + = W n (O Ln /b Ln )/ [ζ pn ] − 1 W n (m Ln /b Ln ) A ¯ n, F ,r + for F . We give these rings structures of Σ -algebras as follows. The ring A¯ n, Ln ,r + and similarly A is considered as a Σ -algebra by using the system {πn }n∈Z0 we chose of p-power roots of π , as in the previous section. On the other hand, using g ∈ G K in Corollary 5.10, put π˜ n = g (πn ) and ζ˜ pn+1 = g (ζ pn+1 ). Then we consider the ring A¯ n, F ,r + as a Σ -algebra by using a system of p-power roots of π containing π˜ n . We define Filr and φr of these rings in the same way as before. Lemma 5.12. The induced ring homomorphism

η¯ : A¯ n,Ln ,r + → A¯ n, F ,r + is a morphism of the category  Mod/Σ . r ,φ

Proof. Firstly, we check that η¯ is Σ -linear. By definition, this homomorphism commutes with the action of the element u ∈ Σ . To show the compatibility with the element Y ∈ Σ , let us consider the commutative diagram

W n (O Ln /b Ln )

¯ n, Ln ,r + A

ηn

η¯

W n (O F /b F )

¯ n, F ,r + , A

η. Note that we have ηn ([πn ]) = [π˜ n ] and ηn ([ζ pn+1 ]) = × ˜ [ζ pn+1 ]. Let a ∈ W ( R ) and v = t / E ([π ]) ∈ W ( R )× be as in the previous section. Let an and v n denote the images of a and v in W n (O Ln /b Ln ), respectively. Then the element v n is a solution of the equation where the horizontal arrows are induced by





E [πn ] v n = 1 + [ζ pn+1 ] + · · · + [ζ pn+1 ] p −1 .

S. Hattori / Journal of Number Theory 129 (2009) 2474–2503

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Similarly, we define elements a˜ n and v˜ n of W n (O F /b F ) using π˜ n and ζ˜ pn+1 . By definition, the element v˜ n is a solution of the equation





E [π˜ n ] v˜ n = 1 + [ζ˜ pn+1 ] + · · · + [ζ˜ pn+1 ] p −1 . Now what we have to show is the equality





 p −1 

η¯ an v n−1 E [πn ]

  p −1 = a˜ n v˜ n−1 E [π˜ n ]

¯ n, F ,r + . Since the element an of W n (O Ln /b Ln ) is a linear combination of the elements in the ring A ¯ n, F ,r + . The elements v˜ n and η¯ ( v n ) satisfy 1, [ζ pn+1 ], . . . , [ζ pn+1 ] p −1 over Z, we have η¯ (an ) = a˜ n in A

¯ n, F ,r + . Since these two elements are invertible, we get η¯ ( v n )−1 E ([π˜ n ]) = the same equation in A v˜ n−1 E ([π˜ n ]) and the equality holds. Since the diagram above is compatible with the Frobenius endomorphisms, we see from the definition that η¯ also preserves Filr and commutes with φr of both sides. 2 Thus we get a homomorphism of abelian groups ∗ ∗ T crys , Ln ,πn ( M ) → T crys, F ,π˜ ( M ). n

Then the following lemma, whose proof is omitted in [3, Subsection 3.13], implies that this homomorphism is an injection. We insert here a proof of this lemma for the convenience of the reader.

¯ n, Ln ,r + → A¯ n, F ,r + is an injection. Lemma 5.13. The ring homomorphism η¯ : A Proof. Let x = (x0 , . . . , xn−1 ) be an element of W n (O Ln /b Ln ) such that





r



η(0) (x0 ), . . . , η(0) (xn−1 ) ∈ [ζ pn ] − 1 W n (m F /b F ),

where η(0) is as in Lemma 5.11. Suppose that x0 = · · · = xm−1 = 0 for some 0  m  n − 1. Let zˆ i ∈ O F be a lift of η(0) (xi ). By Lemma 4.6, we have

r  (0, . . . , 0, zˆ m , . . . , zˆ n−1 ) = [ζ pn ] − 1 ( yˆ 0 , . . . , yˆ n−1 ) for some yˆ 0 , . . . , yˆ n−1 ∈ m F . Thus we get yˆ 0 = · · · = yˆ m−1 = 0 and v K (ˆzm ) > er /( pn−1−m ( p − 1)). Then (n−1−m) Lemma 5.11 implies that xm is contained in the ideal b Ln /b Ln and



r



  x = [ζ pn ] − 1 (0, . . . , 0, y , 0, . . . , 0) + 0, . . . , 0, xm +1 , . . . , xn−1



  ¯ for some y ∈ m Ln /b Ln and xm +1 , . . . , xn−1 ∈ O Ln /b Ln . Repeating this, we see that x is zero in A n, Ln ,r + and the lemma follows. 2 ∗ Now Corollary 4.13 shows that the abelian group T crys , Ln ,πn ( M ) has the same cardinality ∗ ∗ as T ( M ) . This implies that the abelian group T ( M ) has cardinality no less than ¯ crys, F ,π˜

#T ∗

crys, K ,πn

crys, K¯ ,πn

n

( M ). Let g ∈ G K be as in Corollary 5.10. Then we have the following lemma.

Lemma 5.14. The G F n -module T ∗

crys, K¯ ,π˜ n

by the element g.

∗ ( M ) is isomorphic to the conjugate of the G F n -module T crys (M ) , K¯ ,π n

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S. Hattori / Journal of Number Theory 129 (2009) 2474–2503

Proof. Let us consider the composite g Σ → A¯ n,r + − → A¯ n,r +

of the ring homomorphism defined by u → [πn ] and the map induced by g. We can check that this is the natural ring homomorphism defined by u → [π˜ n ] as in the proof of Lemma 5.12. Thus we have an isomorphism of abelian groups

¯ n,r + ) → HomΣ ( M , A¯ n,r + ), HomΣ ( M , A f → g ◦ f ,

¯ n,r + on the right-hand side the filtered φr -module structure over Σ where we consider on the ring A defined by π˜ n . As in the proof of Lemma 5.12, we can check that this isomorphism induces an injection

¯ n,r + ) → HomΣ,Filr ,φ ( M , A¯ n,r + ). HomΣ,Filr ,φr ( M , A r This is also an isomorphism, for the map f → g −1 ◦ f defines its inverse.

2

Thus we have #T ∗

∗ ( M ) = #T crys ( M ). Since Ln is Galois over K , this lemma also shows , K¯ ,πn that the finite Galois extension of F n cut out by the action on T ∗ ¯ ( M ) is L n . Hence we see crys, K ,π˜ crys, K¯ ,π˜ n

n

from Corollary 4.13 that F contains L n and Proposition 5.7 follows. This concludes the proof of Theorem 1.1. 2

Proof of Corollary 1.3. The second assertion follows immediately from Theorem 1.1 and [8, Théorème 1.1]. As for the first assertion, note that if r = 0 then V is unramified and the assertion is trivial. Thus we may assume p  3. Since we have the natural surjection L/ pn L → L/L , we may ˆ ∈ Modr ,φ, N , let us consider the G K -module also assume L = pn L. For M /S ∗ ˆ ˆ ˆ T st ,π (M) = Hom S ,Filr ,φr , N (M, A st ).

ˆ ∈ Mod By [17, Theorem 2.3.5], there exists M /S

r ,φ, N

such that the G K -module L is isomorphic to

∗ (M ˆ ) and the asserˆ ). Then we see that the G K -module L/ pn L is isomorphic to T ∗ (M ˆ / pn M T st ,π st,π tion follows from Theorem 1.1. 2

Remark 5.15. The ramification bound in Theorem 1.1 is sharp for r  1. Indeed, the greatest upper ramification break 1 + e (n + 1/( p − 1)) for r = 1 is obtained by the pn -torsion of the Tate curve K¯ × /π Z (see Remark 5.5). The author does not know whether these bounds are sharp also for r  2. Acknowledgments The author would like to pay his gratitude to Iku Nakamura and Takeshi Saito for their warm encouragements. He wants to thank Manabu Yoshida for kindly allowing him to include the proof of Proposition 5.6. He also wants to thank Ahmed Abbes, Xavier Caruso, Takeshi Tsuji, and especially Victor Abrashkin and Yuichiro Taguchi, for useful discussions and comments. Finally, he is very grateful to the referee for his or her valuable comments to improve this paper. This work was partially supported by 21st Century COE program “Mathematics of Nonlinear Structure via Singularity” at Department of Mathematics, Hokkaido university, and by the JSPS International Training Program (ITP).

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