ON A RAMIFICATION BOUND OF SEMI-STABLE MOD p REPRESENTATIONS OVER A LOCAL FIELD

SHIN HATTORI

Abstract. 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 numbering ramification group G_{K}^{(j)} for j > u(K, r) acts trivially on the mod p representations associated to the semi-stable p-adic G_{K}-representations with Hodge-Tate weights in {0, …, r}, where we put u(K, 1) = 1+e(1+1/(p−1)) and u(K, r) = 1−1/p+e(1+r/(p−1)) for r ≥ 2.

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). Let the maximal ideal of K be denoted by m_{K}, an algebraic closure of K by K̄ and the absolute Galois group of K by G_{K} = Gal(K̄/K). We normalize the valuation v_{K} of K as v_{K}(p) = e and extend this to K̄. Let G_{K}^{(j)} denote the j-th upper numbering ramification group in the sense of [6]. Namely, we put G_{K}^{(j)} = G_{K}^{j−1}, where the latter is the upper numbering ramification group defined in [12].

Let X_{K} be a proper smooth scheme over K and put XK̄ = X_{K} ×_{K} K̄. Consider the r-th p-adic étale cohomology group H_{ét}^{r}(XK̄, ℚ_{p}) and its G_{K}-stable ℤ_{p}-lattices L ⊇ L′. In [6], Fontaine conjectured the upper numbering ramification group G_{K}^{(j)} acts trivially on the G_{K}-module L/L′ for j > e(n + r/(p − 1)) if X_{K} has good reduction and this module is killed by p^{n}. For e = 1 and r < p − 1, this conjecture was solved independently by himself ([7], for n = 1) and Abrashkin ([1], for any n), using the theory of Fontaine-Laffaille ([8]) and the comparison theorem of Fontaine-Messing ([9]) between the p-adic étale cohomology groups of X_{K} and the crystalline cohomology groups of the reduction of X_{K}. From this result, Fontaine also showed some rareness of a proper smooth scheme over ℚ with everywhere good reduction ([7, Théorème 1]). In fact, they proved this ramification bound for the torsion representations of the crystalline p-adic representations of G_{K} with Hodge-Tate weights in {0, …, r} in the case where K is absolutely unramified.

On the other hand, as for a semi-stable p-adic representation V with Hodge-Tate weights in the same range, a similar ramification bound for e = 1 and n = 1 is obtained by Breuil (see [4, Proposition 9.2.2.2]). He showed, assuming Griffiths transversality which in general does not hold, that if e = 1 and r < p − 1, then the ramification group G_{K}^{(j)} acts trivially on the mod p representations of V for j > 2 + 1/(p − 1).

Date: July 7, 2008.

Supported by 21st Century COE Program “Mathematics of Nonlinear Structures via Singularities” at Department of Mathematics, Hokkaido University.

In this paper, we prove a version of the result of Breuil for the case where K is absolutely ramified, under the condition r < p − 1 and n = 1. Our main theorem is the following.

Theorem 1.1. Let r be a non-negative integer such that r < p − 1. Let V be a semi-stable p-adic G_{K}-representation with Hodge-Tate weights in {0, …, r} and L ⊇ L′ be G_{K}-stable ℤ_{p}-lattices in V. Suppose that the quotient L/L′ is killed by p. Then the j-th upper numbering ramification group G_{K}^{(j)} acts trivially on the G_{K}-module L/L′ for j > u(K, r), where we put u(K, 1) = 1 + e(1 + 1/(p − 1)) and u(K, r) = 1 − 1/p + e(1 + r/(p − 1)) for r ≥ 2.

Note that this ramification bound is sharp at least for r = 1, since the upper bound 1+e(1+1/(p−1)) of the ramification is obtained by a mod p representation associated to a Tate curve over K.

For the proof of the theorem, we follow the same lines as in [7]. Thanks to Liu’s theorem ([11]) on G_{K}-stable ℤ_{p}-lattices in semi-stable p-adic representations, it is enough to bound the ramification of the G_{K}-module

T_{st,\underline{π}}^{∗}(M) = Hom_{S,Fil^{r},ϕ_{r},N}(M, Â_{st,∞}),

where M is a p-torsion object of the category \underline{M}^{r} of filtered (ϕ_{r}, N)-modules over S defined by Breuil ([2]) and Â_{st,∞} is a p-adic period ring. Put S̃_{1} = k[u]/(u^{ep}) ≅ S/(p, Fil^{p}S), M̃ = M⊗_{S} S̃_{1}, π_{1} = π^{1/p} and K_{1} = K(π_{1}). We also have a G_{K_{1}}-linear bijection

T_{st,\underline{π}}^{∗}(M)|_{G_{K_{1}}} ≅ T_{crys,π_{1}}^{∗}(M) = Hom_{S̃_{1},Fil^{r},ϕ_{r}}(M̃, OK̄/pOK̄)

with the natural filtered ϕ_{r}-module structure on OK̄/pOK̄. Then the same argument as in [7] gives a ramification bound of the G_{K_{1}}-module T_{crys,π_{1}}^{∗}(M) and careful use of [6, Proposition 1.5] shows the theorem.

Acknowledgments. The author would like to pay his gratitude to Iku Nakamura and Takeshi Saito for their warm encouragements. He also wants to thank Ahmed Abbes, Takeshi Tsuji, Xavier Caruso, Yuichiro Taguchi, Manabu Yoshida and Victor Abrashkin for helpful discussions and comments.

2. Filtered (ϕ_{r}, N)-modules of Breuil

In this section, we recall the theory of filtered (ϕ_{r}, N)-modules of Breuil, which is developed by himself and most recently by Caruso and Liu. 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.

RAMIFICATION BOUND OF SEMI-STABLE MOD p REPRESENTATIONS 3

Let σ be the Frobenius homomorphism of W. We fix once and for all a uniformizer π of K and a system {π_{n}}_{n∈ℤ_{≥0}} of p-power roots of π such that π_{0} = π and π_{n} = π_{n+1}^{p}. Let E(u) be the Eisenstein polynomial of π over W and set S = (W [u]^{PD})^{∧}, where the divided power envelope of W [u] is taken with respect to the ideal (E(u)) and ∧ means the p-adic completion. The ring S is endowed with the σ-semilinear map ϕ : u ↦ u^{p} and the natural filtration induced by the divided power structure. We set ϕ_{t} = p^{−t}ϕ|_{Fil^{t}S} for any non-negative integer t and c = ϕ_{1}(E(u)) ∈ S^{×}. The W -linear derivation N on S is defined by N(u) = −u. We also define a filtration, ϕ, ϕ_{t}, N on S_{n} = S/p^{n}S similarly.

Let r ∈ \{0, …, p − 2\} be an integer. Set ′\underline{M}^{r} to be the category consisting of the following data:

• an S-module M and its S-submodule Fil^{r}M containing Fil^{r}S.M,

• a ϕ-semilinear map ϕ_{r} : Fil^{r}M → M satisfying

ϕ_{r}(s_{r}m) = ϕ_{r}(s_{r})ϕ(m)

for any s_{r} ∈ Fil^{r}S and m ∈ M, where we set ϕ(m) = c^{−r}ϕ_{r}(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(Fil^{r}M) ⊆ Fil^{r}M,

– the following diagram is commutative:

Fil^{r}M ⟶^{ϕ_{r}} M
E(u)N↓  ↓^{cN}
Fil^{r}M ⟶_{ϕ_{r}} M,

and the morphisms of ′\underline{M}^{r} are defined to be the S-linear maps preserving Fil^{r} and commuting with ϕ_{r} and N. The category defined in the same way but dropping the data N is denoted by ′\underline{M}_{0}^{r}. These categories have obvious notions of exact sequences. The category \underline{M}^{r} (resp. \underline{M}_{0}^{r}) is defined as the smallest full subcategory of ′\underline{M}^{r} (resp. ′\underline{M}_{0}^{r}) stable under extensions and containing the objects satisfying the following condition:

• M is free of finite rank over S_{1} and generated as an S_{1}-module by the

image of ϕ_{r}. For p-torsion objects, we have the following categories. Consider the algebra

k[u]/(u^{ep}) ≅ S_{1}/Fil^{p}S_{1} and let this be denoted by S̃_{1}. The algebra S̃_{1} has the natural filtration, ϕ and N induced by those of S. Namely, Fil^{t}S̃_{1} = u^{et}S̃_{1}, ϕ(u) = u^{p} and N(u) = −u. Then the category ′ ˜\underline{M}^{r} consists of the following data:

• an S̃_{1}-module M̃ and its S̃_{1}-submodule Fil^{r}M̃ containing u^{er}M̃,

• a ϕ-semilinear map ϕ_{r} : Fil^{r}M̃ → M̃,

• a k-linear map N : M̃ → M̃ such that – N(sm) = N(s)m + sN(m) for any s ∈ S̃_{1} and m ∈ M̃,

– u^{e}N(Fil^{r}M̃) ⊆ Fil^{r}M̃,

– the following diagram is commutative:

Fil^{r}M̃ ⟶^{ϕ_{r}} M̃
u^{e}N↓  ↓^{cN}
Fil^{r}M̃ ⟶_{ϕ_{r}} M̃,

and the morphisms are defined as before. Its full subcategory ˜ \underline{M}^{r} is defined by the following condition:

• As an S̃_{1}-module, M̃ is free of finite rank and generated by the image of ϕ_{r}.

The categories ′\underline{M̃}_{0}^{r} and \underline{M̃}_{0}^{r} are also defined similarly.

Let D be a weakly admissible filtered (ϕ, N)-module over K satisfying Fil^{0}D_{K} = D_{K} and Fil^{r+1}D_{K} = 0. Set S_{K_{0}} = S ⊗_{W} K_{0} and D = D ⊗_{K_{0}} S_{K_{0}}. Then the S_{K_{0}}-module D is endowed with the natural ϕ-semilinear map ϕ ⊗ σ and K_{0}-linear derivation N ⊗1+1⊗N, which are denoted by ϕ and N, respectively. The filtration on D is defined inductively by Fil^{0}D = D and

Fil^{i+1}D = {x ∈ D| N(x) ∈ Fil^{i}D and f_{π}(x) ∈ Fil^{i+1}D_{K}},

where f_{π} : D → D_{K} is induced by the map S → O_{K} sending u to π. An Ssubmodule M̂ of D is said to be a strongly divisible lattice of D if the following conditions are satisfied:

• the S-module M̂ is free of finite rank,

• M̂ ⊗_{W} K_{0} = D,

• M̂ is stable under ϕ and N,

• ϕ(Fil^{r}M̂) ⊆ p^{r}M̂, where we set Fil^{r}M̂ = M̂ ∩ Fil^{r}D.

We put ϕ_{r} = p^{−r}ϕ|_{Fil_{r} M̂}. Then the S-module M̂ is generated by ϕ_{r}(Fil^{r}M̂) ([2, Proposition 2.1.3]).

Let A_{crys} and Â_{st} be p-adic period rings. These are constructed as follows. Set

R = \underleftarrow{lim}(OK̄/pOK̄ ← OK̄/pOK̄ ← ⋯),

where every arrow is the p-power map. For an element x = (x_{i})_{i∈ℤ_{≥1}} ∈ R, we set

x^{(n)} = lim_{m→∞} x̂_{n+m}^{p^{m}} ∈ O_{ℂ} for n ≥ 0,

where x̂_{i} is a lift of x_{i} in OK̄. The natural ring homomorphism θ is defined by

θ : W (R) → O_{ℂ}
(r_{0}, r_{1}, …) ↦ ∑_{n≥0} p^{n}r_{n}^{(n)}.

Then A_{crys} is the p-adic completion of the divided power envelope of W (R) with respect to Ker(θ) and Â_{st} is the p-adic completion of the divided power polynomial ring A_{crys}⟨X⟩ over A_{crys}. We set A_{crys,∞} = A_{crys} ⊗_{W} K_{0}/W and A_{st,∞} = Â_{st} ⊗_{W} K_{0}/W. Put \underline{π} = (π_{1} mod p, π_{2} mod p, …) ∈ R. These rings are considered as Smodules by the ring homomorphisms S → Â_{st} and Â_{st} → A_{crys} which are defined by u ↦ [\underline{π}]/(1 + X) and X ↦ 0, respectively. The ring A_{crys} is endowed with the natural filtration induced by the divided power structure, the natural Frobenius ϕ and the ϕ-semilinear map ϕ_{t} = p^{−t}ϕ|_{Fil^{t}A_{crys}}. With these structures, A_{crys} and A_{crys,∞} are considered as objects of ′\underline{M}_{0}^{r}. Moreover, the absolute Galois group G_{K} acts naturally on these two rings. As for Â_{st}, its filtration is defined by

Fil^{t}Â_{st} = \{∑_{i≥0} a_{i} \frac{X^{i}}{i!} | a_{i} ∈ Fil^{t−i}A_{crys}, lim_{i→∞}a_{i} = 0\}

and the Frobenius structure of A_{crys} extends to Â_{st} by

ϕ(X) = (1 + X)^{p} − 1,
ϕ_{t} = p^{−t}ϕ|_{Fil_{t}Â_{st}}.

The A_{crys}-linear derivation N on Â_{st} is defined by N(X) = 1 + X. The rings Â_{st} and Â_{st,∞} are objects of ′\underline{M}^{r}. The G_{K}-action on A_{crys} naturally extends to the action on Â_{st}. Indeed, the action of g ∈ G_{K} on Â_{st} is defined by the formula

g(X) = [\underline{ε}(g)](X + 1) − 1,

where g(π_{n}) = ε_{n}(g)π_{n} and \underline{ε}(g) = (ε_{1}(g) mod p, ε_{2}(g) mod p, …) ∈ R.

Put K_{n} = K(π_{n}) and K_{∞} = ∪_{n}K_{n}. For M ∈ \underline{M}^{r}, the G_{K}-module T_{st,\underline{π}}^{∗}(M) and G_{K_{∞}}-module T_{crys,\underline{π}}^{∗}(M) are defined to be

T_{st,\underline{π}}^{∗}(M) = Hom_{S,Fil^{r},ϕ_{r},N}(M, Â_{st,∞}),
T_{crys,\underline{π}}^{∗}(M) = Hom_{S,Fil^{r},ϕ_{r}}(M, A_{crys,∞}).

Then we see as in the proof of [2, Lemme 2.3.1.1] that the natural map

T_{st,\underline{π}}^{∗}(M) → T_{crys,\underline{π}}^{∗}(M)

is bijective and G_{K_{∞}}-linear.

For p-torsion objects in \underline{M}^{r} and \underline{M}_{0}^{r}, we have a simpler description of the functors T_{st,\underline{π}}^{∗} and T_{crys,\underline{π}}^{∗} ([5]). We consider Â = (OK̄/pOK̄)⟨X⟩ as an object of ′ ˜\underline{M}^{r} by the natural surjection

Â_{st}/pÂ_{st} → Â_{st}/(p, Fil^{p}A_{crys}) ≅ Â,

where the last isomorphism is defined by X ↦ X. This surjection also induces the natural G_{K}-action on Â. Similarly, the algebra OK̄/pOK̄ is considered as an object of ′\underline{M̃}_{0}^{r}. The filtration of OK̄/pOK̄ is given by

Fil^{r}(OK̄/pOK̄) = \{x̂ ∈ OK̄| v_{K}(x̂) ≥ \frac{er}{p} \}/pOK̄

and the Frobenius structure by

ϕ_{r}(x) = \frac{x̂^{p}}{(−p)^{r}} mod p,

where x̂ denotes a lift of x in OK̄. For a p-torsion object M ∈ \underline{M}^{r}, we set T (M) = M ⊗_{S} S̃_{1}. This S̃_{1}-module is naturally considered as an object of ˜ \underline{M}^{r}. By [5, Lemme 2.3.4], we have a G_{K}-linear isomorphism

T_{st,\underline{π}}^{∗}(M) → Hom_{S̃_{1},Fil^{r},ϕ_{r},N}(T (M), Â).

Similarly, for a p-torsion object M ∈ \underline{M}_{0}^{r}, the S̃_{1}-module T (M) = M ⊗_{S} S̃_{1} has a natural structure as an object of \underline{M̃}_{0}^{r}. Then we have a G_{K_{∞}}-linear isomorphism

T_{crys,\underline{π}}^{∗}(M) → Hom_{S̃_{1},Fil^{r},ϕ_{r}}(T (M), OK̄/pOK̄),

where the module on the right-hand side is in fact a G_{K_{1}}-module. By definition, the action of g ∈ G_{K} on Â is defined by

g(X) = ε_{1}(g)(1 + X) − 1.

Thus we see that, for a p-torsion object M ∈ \underline{M}^{r}, the natural map

Hom_{S̃_{1},Fil^{r},ϕ_{r},N}(T (M), Â) → Hom_{S̃_{1},Fil^{r},ϕ_{r}}(T (M), OK̄/pOK̄)

is bijective and G_{K_{1}}-linear. Both sides are independent of the choice of π_{n} (n ≥ 2). We refer these modules as T_{st,π_{1}}^{∗}(M) and T_{crys,π_{1}}^{∗}(M).

Finally, let D and D be as above and M̂ be a strongly divisible lattice in D. Then the G_{K}-module

T̂_{st,\underline{π}}^{∗}(M̂) = Hom_{S,Fil^{r},ϕ_{r},N}(M̂, Â_{st})

is naturally considered as a G_{K}-stable ℤ_{p}-lattice in V_{st}^{∗}(D). By Liu’s theorem ([11, Theorem 2.3.5]), the functor T̂_{st,\underline{π}}^{∗} gives an anti-equivalence of categories between the category of strongly divisible lattices in D and the category of G_{K}-stable ℤ_{p}- lattices in V_{st}^{∗}(D). The S-module M = M̂/p^{n}M̂ has a natural structure as an object of \underline{M}^{r}. There is an exact sequence of G_{K}-modules

0 ⟶ T̂_{st,\underline{π}}^{∗}(M̂) ⟶^{p^{n}} T̂_{st,\underline{π}}^{∗}(M̂) ⟶ T_{st,\underline{π}}^{∗}(M) ⟶ 0.

3. Galois action on T_{crys,π_{1}}^{∗}(M) over K_{1}

Let us fix a p-th root π_{1} of π and set K_{1} = K(π_{1}), as before. For an algebraic extension F of K, we put

𝔟_{F} = \{x ∈ O_{F} | v_{K}(x) > \frac{er}{p − 1}\}.

Let E be an algebraic extension of K_{1}. We define on the O_{K_{1}}/pO_{K_{1}}-algebra O_{E}/𝔟_{E} a structure of a filtered ϕ_{r}-module over S̃_{1}, as follows. We consider the algebra O_{E}/𝔟_{E} as an S̃_{1}-algebra by u ↦ π_{1}. Define an S̃_{1}-submodule Fil^{r}(O_{E}/𝔟_{E}) of O_{E}/𝔟_{E} by

Fil^{r}(O_{E}/𝔟_{E}) = u^{er}(O_{E}/𝔟_{E}) = \{x̂ ∈ O_{E} | v_{K}(x̂) ≥ \frac{er}{p} \}/𝔟_{E}

and a ϕ-semilinear map ϕ_{r} by

ϕ_{r} : Fil^{r}(O_{E}/𝔟_{E}) → O_{E}/𝔟_{E}
x ↦ \frac{x̂^{p}}{(−p)^{r}} mod 𝔟_{E},

where x̂ is a lift of x in O_{E}. We see that ϕ_{r} is independent of the choice of a lift x̂ and ϕ-semilinear.

Let M be a p-torsion object in \underline{M}^{r}. Put M̃ = T (M) and

T_{crys,π_{1},E}^{∗}(M) = Hom_{S̃_{1},Fil^{r},ϕ_{r}}(M̃, O_{E}/𝔟_{E}).

For finite extensions E ⊆ E′ of K_{1}, we have a natural injection of filtered ϕ_{r}- modules over S̃_{1}

O_{E}/𝔟_{E} → O_{E′}/𝔟_{E′}

and this induces an injection of abelian groups

T_{crys,π_{1},E}^{∗}(M) → T_{crys,π_{1},E′}^{∗}(M).

Thus we have a natural identification of abelian groups

T_{crys,π_{1},K̄}^{∗}(M) = ⋃_{E/K_{1}: finite}T_{crys,π_{1},E}^{∗}(M).

Take an adapted basis e_{1}, …, e_{d} of M̃ ([2, Proposition 2.2.1.3]) such that

M̃ = S̃_{1}e_{1} ⊕ ⋯ ⊕ S̃_{1}e_{d}
Fil^{r}M̃ = u^{r_{1}} S̃_{1}e_{1} ⊕ ⋯ ⊕ u^{r_{d}} S̃_{1}e_{d}

with some integers r_{1}, …, r_{d} satisfying 0 ≤ r_{i} ≤ er for any i. Define G(u) ∈ GL_{d}(S̃_{1}) by

ϕ_{r} ( u^{r_{1}}e_{1} ; ⋮ ; u^{r_{d}}e_{d} ) = G(u) ( e_{1} ; ⋮ ; e_{d} )

and choose its lift Ĝ(u) ∈ GL_{d}(W [[u]]). For an ideal I ⊆ 𝔟_{E}, consider the equation in O_{E}/I

(1)  ( \frac{π^{r_{1}}}{(−p)^{r}} x̂_{1}^{p} ; ⋮ ; \frac{π^{r_{d}}}{(−p)^{r}} x̂_{d}^{p} ) mod I = Ĝ(π_{1}) ( x_{1} ; ⋮ ; x_{d} ) , with x_{i} ∈ π_{1}^{er−r_{i}}(O_{E}/I).

Here x̂_{i} denotes a lift of x_{i} in O_{E} and this equation is independent of the choice of these lifts. The elements of

T_{crys,π_{1},E}^{∗}(M) = Hom_{S̃_{1},Fil^{r},ϕ_{r}}(M̃, O_{E}/𝔟_{E})

correspond bijectively to the solutions of this equation for I = 𝔟_{E}.

Lemma 3.1. Let E be a finite extension of K_{1} and l > e(E)r/(p−1). Then, every solution (x_{1}, …, x_{d}) of the equation (1) for I = m_{E}^{l} lifts uniquely to a solution of this equation for I = m_{E}^{l+1}.

Proof. Let (x_{1}, …, x_{d}) be such a solution and take a lift x̂_{i} of x_{i} in O_{E}. Let π_{E} denote a uniformizer of E. Then, for some c_{1}, …, c_{d} ∈ O_{E}, we have

( \frac{π^{r_{1}}}{(−p)^{r}} x̂_{1}^{p} ; ⋮ ; \frac{π^{r_{d}}}{(−p)^{r}} x̂_{d}^{p} ) = Ĝ(π_{1}) ( x̂_{1} ; ⋮ ; x̂_{d} ) + π_{E}^{l} ( c_{1} ; ⋮ ; c_{d} ) .

For ŷ_{1}, …, ŷ_{d} ∈ O_{E}, consider the equation

( \frac{π^{r_{1}}}{(−p)^{r}} (x̂_{1} + π_{E}^{l}ŷ_{1})^{p} ; ⋮ ; \frac{π^{r_{d}}}{(−p)^{r}} (x̂_{d} + π_{E}^{l}ŷ_{d})^{p} ) ≡ Ĝ(π_{1}) ( x̂_{1} + π_{E}^{l}ŷ_{1} ; ⋮ ; x̂_{d} + π_{E}^{l}ŷ_{d} ) mod π_{E}^{l+1}.

This is equivalent to

( ŷ_{1} ; ⋮ ; ŷ_{d} ) ≡ Ĝ(π_{1})^{−1} \{( c_{1} ; ⋮ ; c_{d} ) + ( \frac{π^{r_{1}}}{(−p)^{r}} (∑_{t=1}^{p} \binom{p}{t}x̂_{1}^{p−t}π_{E}^{l(t−1)}ŷ_{1}^{t}) ; ⋮ ; \frac{π^{r_{d}}}{(−p)^{r}} (∑_{t=1}^{p} \binom{p}{t}x̂_{d}^{p−t}π_{E}^{l(t−1)}ŷ_{d}^{t}) )\} mod π_{E},

where \binom{p}{t} denotes the binomial coefficient p!/(t!(p − t)!).

We claim that every entry of the last term of this equation is a polynomial of ŷ_{1}, …, ŷ_{d} whose coefficients are in the maximal ideal m_{E}. Indeed, the coefficient of ŷ_{i}^{p}

\frac{π^{r_{i}}}{(−p)^{r}} π_{E}^{l(p−1)}

has the valuation

e(E/K)r_{i} − e(E)r + l(p − 1) > 0.

Similarly, the coefficient of ŷ_{i}^{t} (1 ≤ t ≤ p − 1)

\frac{π^{r_{i}}}{(−p)^{r}} \binom{p}{t}x̂_{i}^{p−t}π_{E}^{l(t−1)}

has the valuation

e(E/K)r_{i} − e(E)(r − 1) + (p − t)v_{E}(x̂_{i}) + l(t − 1)
= e(E/K)r_{i} − e(E)(r − 1) + pv_{E}(x̂_{i}) − l + t(l − v_{E}(x̂_{i})).

If l ≥ v_{E}(x̂_{i}), then the minimum of this value for t = 1, …, p − 1 is obtained by t = 1. This minimum value is equal to

e(E/K)r_{i} − e(E)(r − 1) + (p − 1)v_{E}(x̂_{i})

≥ \frac{e(E)(p − r)}{p} + \frac{e(E/K)r_{i}}{p} > 0.

If l < v_{E}(x̂_{i}), then the minimum is obtained by t = p − 1 and is equal to

e(E/K)r_{i} − e(E)(r − 1) + (p − 2)l + v_{E}(x̂_{i})

> e(E/K)r_{i} − e(E)(r − 1) + \frac{e(E)r(p − 2)}{p − 1} + v_{E}(x̂_{i})

= e(E/K)r_{i} + e(E)(1 − \frac{r}{p − 1}) + v_{E}(x̂_{i}) > 0.

Thus we can solve uniquely this equation. □

Corollary 3.2. There is a G_{K_{1}}-linear isomorphism

T_{crys,π_{1}}^{∗}(M) ≅ T_{crys,π_{1},K̄}^{∗}(M).

In particular, we have #T_{crys,π_{1},K̄}^{∗}(M) = p^{d} for d = dim_{S_{1}} M.

Proof. Applying Lemma 3.1 for a sufficiently large finite extension E of K_{1}, we see that the natural map

T_{crys,π_{1}}^{∗}(M) → T_{crys,π_{1},K̄}^{∗}(M)

induced by OK̄/pOK̄ → OK̄/𝔟K̄ is bijective. The last assertion follows from [2, Lemme 2.3.1.2]. □

Lemma 3.3. The fixed part T_{crys,π_{1},K̄}^{∗}(M)^{G_{E}} is equal to T_{crys,π_{1},E}^{∗}(M).

Proof. From Lemma 3.1, we see that the elements of T_{crys,π_{1},K̄}^{∗}(M) correspond bijectively to the solutions of the equation (1) for I = 0 in OK̄. The uniqueness of the lift shows that g ∈ G_{K_{1}} fixes a solution in OK̄ if and only if g fixes its image in OK̄/𝔟K̄. This concludes the proof. □

Corollary 3.4. Let L_{1} be the finite Galois extension of K_{1} corresponding to the kernel of the map G_{K_{1}} → Aut(T_{crys,π_{1}}^{∗}(M)). Then an algebraic extension E of K_{1} contains L_{1} if and only if #T_{crys,π_{1},E}^{∗}(M) = p^{d}, where d = dim_{S_{1}} M.

Proof. An algebraic extension E of K_{1} contains L_{1} if and only if the action of G_{E} on T_{crys,π_{1},K̄}^{∗}(M) is trivial. By Lemma 3.3, this is equivalent to T_{crys,π_{1},K̄}^{∗}(M) = T_{crys,π_{1},E}^{∗}(M). Then the corollary follows. □

4. Ramification bound

In this section, we prove Theorem 1.1. Take G_{K}-stable ℤ_{p}-lattices L ⊇ L′ in V such that L′ ⊇ pL. Since the G_{K}-module L/L′ is a quotient of L/pL, we may assume L′ = pL. For r = 0, we see that the G_{K}-module V is unramified and the theorem is trivial. Thus we may assume r > 0. Then, by Liu’s theorem ([11, Theorem 2.3.5]), it suffices to show the following.

Theorem 4.1. Let r > 0 be an integer with r < p − 1. For M ∈ \underline{M}^{r} which is killed by p, G_{K}^{(j)} acts trivially on T_{st,π_{1}}^{∗}(M) for j > u(K, r).

Let L and L_{1} be the finite Galois extensions of K and K_{1} corresponding to the kernels of the maps

G_{K} → Aut(T_{st,π_{1}}^{∗}(M)) and G_{K_{1}} → Aut(T_{st,π_{1}}^{∗}(M)),

respectively. Similarly, put K̃_{1} = K_{1}(ζ_{p}) and let L̃_{1} denote the finite extension of K̃_{1} corresponding to the kernel of the map

GK̃_{1} → Aut(T_{st,π_{1}}^{∗}(M)).

Then we have L_{1} = LK_{1} and L̃_{1} = LK̃_{1} = L_{1}K̃_{1}, where the latter is a Galois extension of K. In the following, we bound the ramification of L̃_{1} over K.

For an algebraic extension E of K, we put

𝔞_{E/K}^{m} = {x ∈ O_{E} | v_{K}(x) ≥ m}.

Proposition 4.2. Let E be an algebraic extension of K. If m > u(K, r) and there exists an O_{K}-algebra homomorphism

η : OL̃_{1} → O_{E}/𝔞_{E/K}^{m},

then there exists a K-algebra injection L̃_{1} → E.

Proof. We mimic the proof of [7, Théorème 2]. By assumption, we have m > er/(p − 1) and 𝔟_{E} ⊇ 𝔞_{E/K}^{m}. Thus η induces an O_{K}-algebra homomorphism

OL̃_{1} → O_{E}/𝔟_{E}.

Note that K̃_{1} = K_{1}(ζ_{p}) is a Galois extension of K and its greatest upper ramification break u_{K̃_{1}/K} ([6]) is

u_{K̃_{1}/K} = 1 + e(1 + \frac{1}{p − 1}) < m.

As η induces an O_{K}-algebra homomorphism OK̃_{1} → O_{E}/𝔞_{E/K}^{m}, from [6, Proposition 1.5] we get a K-linear injection K̃_{1} → E. Thus we see that E contains π_{1} and ζ_{p}. More precisely, we have the following lemma.

Lemma 4.3. η(π_{1}) ≡ π_{1}ζ_{p}^{i} mod 𝔟_{E} for some i.

Proof. As η is O_{K}-linear, the equality η(π_{1})^{p} = π holds in O_{E}/𝔞_{E/K}^{m}. Set x̂ to be a lift of η(π_{1}) in O_{E}. Then we have

v_{K}(x̂^{p} − π) = ∑_{i=0}^{p−1} v_{K}(x̂ − π_{1}ζ_{p}^{i}) ≥ m.

Take i such that v_{K}(x̂ − π_{1}ζ_{p}^{i}) ≥ v_{K}(x̂ − π_{1}ζ_{p}^{i′}) holds for any i′. From the equality

v_{K}(x̂ − π_{1}ζ_{p}^{i}) = v_{K}(x̂ − π_{1}ζ_{p}^{i′} + π_{1}(ζ_{p}^{i′} − ζ_{p}^{i})),

we see that v_{K}(x̂ − π_{1}ζ_{p}^{i′}) ≤ 1/p + e/(p − 1) and

v_{K}(x̂ − π_{1}ζ_{p}^{i}) ≥ m − e − \frac{p − 1}{p}.

Since

m − e − \frac{p − 1}{p} > \frac{er}{p − 1}

by assumption, we have x̂ ≡ π_{1}ζ_{p}^{i} mod 𝔟_{E} and the lemma follows. □

Lemma 4.4. The O_{K}-algebra homomorphism η induces an O_{K}-algebra injection

η̄: OL̃_{1}/𝔟L̃_{1} → O_{E}/𝔟_{E}.

Proof. We write the Eisenstein polynomial of a uniformizer πL̃_{1} of L̃_{1} over O_{K} as

P (T) = T ẽ + c_{1}T^{ẽ−1} + ⋯ + c_{ẽ−1}T + cẽ,

where ẽ = e(L̃_{1}/K). Then t = η(πL̃_{1}) satisfies

tẽ = −(c_{1}t^{ẽ−1} + ⋯ + c_{ẽ−1}t + cẽ)

in O_{E}/𝔞_{E/K}^{m}. Let t̂ be a lift of t in O_{E}. Since m > 1, we have v_{E}(t̂) = e(E/K)/ẽ. The condition n > e(L̃_{1})r/(p − 1) is equivalent to the condition

v_{E}(t̂^{n}) > \frac{e(L̃_{1})r}{p − 1} · \frac{e(E/K)}{ẽ} = \frac{e(E)r}{p − 1} .

Thus the lemma follows. □

We give the O_{K}-algebras OL̃_{1}/𝔟L̃_{1} and O_{E}/𝔟_{E} natural structures of filtered ϕ_{r}- modules over S̃_{1} as follows. The O_{K}-algebra OL̃_{1}/𝔟L̃_{1} is considered as an S̃_{1}-module by u ↦ π_{1}. Put

Fil^{r}(OL̃_{1}/𝔟L̃_{1}) = \{x̂ ∈ OL̃_{1} | v_{K}(x̂) ≥ \frac{er}{p} \}/𝔟L̃_{1}

and for x ∈ Fil^{r}(OL̃_{1}/𝔟L̃_{1}), set

ϕ_{r}(x) = \frac{x̂^{p}}{(−p)^{r}} mod 𝔟L̃_{1},

as in Section 3. On the other hand, we consider O_{E}/𝔟_{E} as an S̃_{1}-module by u ↦ π_{1}ζ_{p}^{i} and define Fil^{r} and ϕ_{r} of O_{E}/𝔟_{E} in the same way as OL̃_{1}/𝔟L̃_{1}. From the definition, we see that η̄ is S̃_{1}-linear. We can also check that η̄ preserves Fil^{r} and ϕ_{r} of both sides, just as in the proof of [7, Théorème 2].

Then the injection η̄ induces an injection of abelian groups

T_{crys,L̃_{1},π_{1}}^{∗}(M) → T_{crys,E,π_{1}ζ_{p}^{i}}^{∗}(M).

Since L̃_{1} contains L_{1}, the abelian group T_{crys,L̃_{1},π_{1}}^{∗}(M) is of order p^{d} by Corollary 3.4, where d = dim_{S_{1}} M. This implies #T_{crys,E,π_{1}ζ_{p}^{i}}^{∗}(M) = p^{d}. Let L′_{1} denote the finite Galois extension of K′_{1} = K(π_{1}ζ_{p}^{i}) corresponding to the kernel of the map

G_{K′_{1}} → Aut(T_{crys,E,π_{1}ζ_{p}^{i}}^{∗}(M)).

Then, again by Corollary 3.4, we see that E ⊇ L′_{1}. However, by definition, the finite extension L′_{1} is the conjugate field g(L_{1}) of L_{1} over K by some element g ∈ G_{K}. Since L̃_{1} = L_{1}K̃_{1} is Galois over K, the Galois group G_{L′_{1}K̃_{1}} = G_{L′_{1}} ∩ GK̃_{1} is equal to

gG_{L_{1}}g^{−1} ∩ GK̃_{1} = g(G_{L_{1}} ∩ GK̃_{1})g^{−1} = gGL̃_{1}g^{−1} = GL̃_{1}.

Hence L̃_{1} is equal to L′_{1}K̃_{1}. Since E contains K̃_{1}, we conclude that E also contains L̃_{1}. □

Corollary 4.5. we have the inequality

u_{L/K} ≤ u_{L̃_{1}/K} ≤ u(K, r).

Proof. This follows from [10, Proposition 5.6]. □

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E-mail address: shin-h@math.sci.hokudai.ac.jp

Department of Mathematics, Hokkaido University