HASSE-ARF THEOREM FOR 𝔽_{p}-VECTOR SPACE SCHEMES OF RANK TWO

SHIN HATTORI

1. Introduction

Let p be an odd prime and f be an elliptic modular form of level N prime to p and weight k ≤ p + 1. Let us consider its associated mod p Galois representation ρ̄_{f} : G_{ℚ} → GL_{2}(𝔽̄_{p}) and its restriction to the inertia subgroup I_{p}. The theorem of Deligne and Fontaine asserts that the tame characters appearing in ρ̄_{f}|_{I_{p}} are determined by k.

Theorem 1.1 (Deligne, Fontaine).

ρ̄_{f}|_{I_{p}} = \{ \begin{matrix}(\begin{matrix}χ_{p}^{k−1} & ∗ \\ 0 & 1 \end{matrix}) & if f is ordinary at p, \\ (\begin{matrix}θ_{p^{2}−1}^{k−1} & 0 \\ 0 & θ_{p^{2}−1}^{p(k−1)} \end{matrix}) & if f is supersingular at p, \end{matrix}

where χ_{p} is the mod p cyclotomic character and θ_{d} is the fundamental character of level d in the sense of [14].

This classification is the basis for the local analysis of ρ̄_{f}, especially for the Serre conjecture of mod p modular forms ([14]). We have two proofs of this theorem for k < p: one uses Raynaud’s full faithful theorem for finite flat representations ([9, section 6]) and the other uses p-adic Hodge theory and the Fontaine-Laffaille functor ([8, Proposition 4.1.1]). In both proofs, it is crucial that p is absolutely unramified, and this is the very obstacle to carry out a similar analysis on the weight of a modular form and its mod p Galois representation for a totally real number field F. In this note, we propose a new approach to tackle this problem which is applicable without any restriction to the ramification index e of F at p, at least in the case of parallel weight (2, …, 2). Namely, we prove the following conjecture in the reducible case for 𝔽 = 𝔽_{p}.

Conjecture 1.2. Let K be a complete discrete valuation field of mixed characteristic (0, p) with perfect residue field and I_{K} be its inertia subgroup. Let 𝔽 be a finite extension of 𝔽_{p} and G be a finite flat 𝔽-vector space scheme of rank 2 over O_{K}. Write c = c(G) for the conductor of G ([2], [3]) and k/l for the prime-to-p part of c mod ℤ. Then an I_{K}-module G(K̄) ⊗_{𝔽} 𝔽̄_{p} contains θ_{l}^{k} as an I_{K}-submodule. Moreover, if G(K̄) is reducible, then we have θ_{l}^{k} ⊆ G(K̄).

Date: May 1, 2007.

This conjecture can be regarded as the counterpart for finite flat group schemes of the Hasse-Arf theorem in the classical ramification theory. In fact, if the Galois group G_{K} acts trivially on G(K̄), this is equivalent to the assertion that, for a complete discrete valuation field M and an abelian extension L of M whose integer ring is a G-torsor over O_{M}, the denominator of the conductor c(L/M) is p-power. In this case, the assertion follows easily from the theorem of Herbrand for finite flat group schemes ([1, Lemme 2.10]).

To prove the conjecture for 𝔽 = 𝔽_{p}, we will firstly show the compatibility of the theory of Breuil ([5]) with respect to a base extension in K_{∞} = K(π^{p^{−∞}}). This makes us possible to describe a defining equation of G explicitly. By virtue of the full faithful theorem of Breuil ([6, Theorem 3.4.3]), such a base change is harmless to study finite flat representations. Next we gather some elementary lemmas for the calculation of the conductor. As a corollary, we determine the conductor of a Raynaud 𝔽-vector space scheme, which is independent of the proof of the main theorem. Then we prove the main theorem by a lengthy calculation. In the forthcoming paper [12], we prove the conjecture in general, by a more geometrical method.

2. Base change property for a filtered ϕ_{1}-module of Breuil

In this section, we briefly recall the theory of a filtered ϕ_{1}-module of Breuil ([5]) and give a proof of its compatibility with the base change from K to K_{∞}.

Let K be a complete discrete valuation field of mixed characteristic (0, p), k be its residue field which we suppose to be perfect in this section, e be its absolute ramification index, W = W (k) and σ be the Frobenius of W. We fix once and for all an uniformizer π of K. Let E(u) = u^{e} − pF (u) be the Eisenstein polynomial of π over W and Set S = S_{π} = (W [u]^{PD})^{∧}, where the divided power envelope of W [u] is taken with respect to an ideal (E(u)) and compatibility with the natural divided power structure on pW, and ∧ means the π-adic completion. The ring S is endowed with a σ-semilinear map ϕ : u ↦ u^{p}, which we also call Frobenius, and the natural filtration induced by the divided power structure. We set ϕ_{1} = 1/p.ϕ|_{Fil^{1}S} and c = ϕ_{1}(E(u)) ∈ S^{×}. We define ϕ, ϕ_{1} and a filtration on S_{n} = S/p^{n} similarly.

In [5], following categories of filtered ϕ_{1}-modules are defined. Set ′M be a category consisting of following data;

• an S-module M and its S-submodule Fil^{1}M containing Fil^{1}SM,

• ϕ-semilinear map ϕ_{1} : Fil^{1}M → M satisfying ϕ_{1}(s_{1}m) = ϕ_{1}(s_{1})ϕ(m),

where s_{1} ∈ Fil^{1}S, m ∈ M and ϕ(m) = c^{−1}ϕ_{1}(E(u)m). Let M_{1} be a full subcategory of ′M consisting of M satisfying

• the S_{1}-module M is free of finite rank,

• ϕ_{1}(Fil^{1}M) generates M as an S-module. and M be the minimal full subcategory of ′M which contains M_{1} and stable under extension.

The category M is shown to be categorically anti-equivalent to the category (p-Gr/O_{K}) of the finite flat group schemes over O_{K} which is killed by some p-power ([5]). Let us recall the definition of this equivalence. Let Spf(O_{K})_{syn} be the category of the p-adic formal schemes of formally syntomic, endowed with the Grothendieck topology generated by the surjective families of formally syntomic morphisms. Write (Ab/O_{K}) for the category of the abelian sheaves on Spf(O_{K})_{syn}. The sheaf O_{n,π} and J_{n,π} is defined by the formula

O_{n,π}(𝔛) = H_{crys}^{0}((𝔛_{n}/S_{n})_{crys}, O_{𝔛_{n}/S_{n}})

and

J_{n,π}(𝔛) = H_{crys}^{0}((𝔛_{n}/S_{n})_{crys}, J_{𝔛_{n}/S_{n}}),

where 𝔛_{n} = 𝔛/p^{n}. We also set O_{∞,π} = \underrightarrow{lim} O_{n,π} and J_{∞,π} = \underrightarrow{lim} J_{n,π}. We denote by ϕ : O_{n,π} → O_{n,π} the crystalline Frobenius map. We can define the natural morphism ϕ_{1} : J_{n,π} → O_{n,π} which makes the following diagram commutative

J_{n,π} ⟶^{ϕ_{1}} O_{n,π}
↑  ↓^{×p}
J_{n+1,π} ⟶^{ϕ} O_{n+1,π}.

Let G ∈ (p-Gr/O_{K}) and M ∈ M. Define

Mod_{K}(G) = Hom_{(Ab/O_{K})}(G, O_{∞,π})

and

Gr_{K}(M) = Hom_{′M}(M, O_{∞,π}).

Then the main theorem of [5] is the following.

Theorem 2.1 (Breuil). The functor Gr_{K} defines an anti-equivalence of categories M → (p-Gr/O_{K}) and its quasi-inverse is Mod_{K}.

Next we consider the base change theorem of the functor Gr for an extension K_{1} = K(π^{1/p}) over K. This extension is totally ramified of degree p. The minimal polynomial of π_{1} = π^{1/p} over W is E_{1}(v) = E(v^{p}) = v^{ep} − pF (v^{p}). Set S′ = S_{π_{1}} = (W [v]^{P D})^{∧}, where the divided power envelope is taken with respect to (E_{1}(v)) and compatibility with the natural divided power structure on pW. The ring S′ has a σsemilinear endomorphism ϕ : S′ → S′ defined by v ↦ v^{p} and a ϕsemilinear map Fil^{1}S′ → S′ satisfying ϕ |_{Fil^{1}S′}= pϕ_{1}. We have a ring homomorphism S → S′ which maps u to v^{p}. This respects the filtration and ϕ_{1}.

Lemma 2.2. The S-module S′ is free of finite rank.

Proof. The W [u]-algebra W [v] is free of finite rank. We have (E(u))W [v] = (E_{1}(v)). Therefore W [v]^{P D} = W [u]^{P D} ⊗_{W [u]} W [v] from [4, Proposition 3.21] and W [u]^{P D} → W [v]^{P D} is also free of finite rank. Thus (W [v]^{P D})^{∧} = (W [u]^{P D})^{∧} ⊗_{W [u]^{P D}} W [v]^{P D}. This concludes the proof.

□

Let us denote the category of filtered ϕ_{1}-modules over S′ by ′M′ and M′. From the lemma above, we can define a filtered ϕ_{1}-module structure on M′ = M ⊗_{S} S′ for any M ∈ ′M by Fil^{1}M′ = (Fil^{1}M)⊗_{S} S′ and ϕ_{1,M′} = ϕ_{1} ⊗ ϕ. If M ∈ M, then we have M′ ∈ M′.

For a presheaf F on Spf(O_{K})_{syn}, we denote by F|_{O_{K_{1}}} the restriction of F to Spf(O_{K_{1}})_{syn}. If F is a sheaf on Spf(O_{K})_{syn}, then F|_{O_{K_{1}}} is also a sheaf on Spf(O_{K_{1}})_{syn}.

Define a morphism Ψ_{M} : Gr(M)|_{O_{K_{1}}} → Gr(M′) of (Ab/O_{K_{1}}) as follows. For any 𝔛′, formally syntomic over Spf(O_{K_{1}}), we want to set Ψ_{M,𝔛′} : Hom_{′M}^{S}(M, O_{n,π}(𝔛′)) → Hom_{′M′}^{S′}(M ⊗_{S} S′, O_{n,π_{1}}(𝔛′)) by f ↦ (m ⊗ s′ ↦ s′.pr_{𝔛′}^{∗}(f(m))), where pr_{𝔛′}^{∗} : O_{n,π}(𝔛′) = H_{crys}^{0}(𝔛′_{n}/S_{n}) → H_{crys}^{0}(𝔛′_{n}/Sn^{′}) = O_{n,π_{1}}(𝔛′) is the natural pull-back. The map pr_{𝔛′}^{∗} respects the filtration. We have to show the compatibility with ϕ_{1}.

Consider 𝔛′ = Spf(𝔄′). We can write 𝔄′ = O_{K_{1}}⟨X′_{1}, …, X′_{r}⟩/(f_{1}, …, f_{s}), where O_{K_{1}}⟨X′_{1}, …, X′_{r}⟩ = O_{K_{1}}[X′_{1}, …, X′_{r}]^{∧} and f_{1}, …, f_{s} is a regular sequence in that ring ([5, Lemme 2.2.1]). Put 𝔄i^{′} = O_{K_{1}}⟨X′_{0}^{p^{−i}}, …, X′_{r}^{p^{−i}}⟩/(X′_{0}− π_{1}, f_{1}, …, f_{s}) and 𝔄∞^{′} = \underrightarrow{lim} 𝔄i^{′}. The W -algebra 𝔄i^{′} is isomorphic to

O_{K}[T]/(T^{p} − π)⟨X′_{0}^{p^{−i}}, …, X′_{r}^{p^{−i}}⟩/(X′_{0} − T, f_{1}, …, f_{s})

= W [u, T]/(E(u), T^{p} − u)⟨X′_{0}^{p^{−i}}, …, X′_{r}^{p^{−i}}⟩/(X′_{0} − T, f_{1}, …, f_{s})

= W ⟨X′_{0}^{p^{−i}}, …, X′_{r}^{p^{−i}}⟩/(E(X′_{0}^{p}), f_{1}, …, f_{s}).

We also set A∞^{′} = 𝔄∞^{′}/p = k[X′_{0}^{p^{−∞}}, …, X′_{r}^{p^{−∞}}]/(X′_{0}^{ep}, f̄_{1}, …, f̄_{s}).

Note that the formal scheme Spf(𝔄i^{′}) is a covering of Spf(𝔄′) in Spf(O_{K_{1}})_{syn}.

Lemma 2.3. The following two sequences are exact;

0 → O_{r,π}(𝔄′_{∞}) →^{×p^{s}} O_{r+s,π}(𝔄∞^{′}) → O_{s,π}(𝔄′_{∞}) → 0

0 → J_{r,π}(𝔄′_{∞}) →^{×p^{s}} J_{r+s,π}(𝔄∞^{′}) → J_{s,π}(𝔄′_{∞}) → 0.

In particular, there are exact sequences in (Ab/O_{K_{1}})

0 → O_{r,π}|_{O_{K_{1}}} →^{×p^{s}} O_{r+s,π}|_{O_{K_{1}}} → O_{s,π}|_{O_{K_{1}}} → 0,

and

0 → J_{r,π}|_{O_{K_{1}}} →^{×p^{s}} J_{r+s,π}|_{O_{K_{1}}} → J_{s,π}|_{O_{K_{1}}} → 0.

Proof. We repeat just the same argument as [5, Lemme 2.3.2].

Note that O_{n,π}(𝔄′_{∞}) = H_{crys}^{0}(𝔄′_{∞}/p^{n}/S_{n}) is isomorphic to (W_{n}(A′_{∞}) ⊗_{W_{n},σ^{n}} W_{n}[u])^{P D}, where the divided power envelope in the right hand side is taken with respect to the kernel of a surjection W_{n}(A′_{∞}) ⊗_{W_{n},σ^{n}} W_{n}[u] → 𝔄′_{∞}/p^{n} which sends (x_{0}, …, x_{n−1}) ⊗ u to X′_{0}^{p} ∑_{k=0}^{n−1} p^{k}x̂_{k}^{p^{n−k}}, and compatibility with the natural divided power structure on pW. Here we denote an lifting of x_{k} in 𝔄′_{∞}/p^{n} by x̂_{k}. In fact, this surjection induces a thickening (W_{n}(A′_{∞}) ⊗_{W_{n},σ^{n}} W_{n}[u])^{P D} → 𝔄′_{∞}/p^{n} of 𝔄′_{∞}/p^{n} over S_{n} and thus we have the natural projection H_{crys}^{0}(𝔄′_{∞}/p^{n}/S_{n}) → (W_{n}(A′_{∞}) ⊗_{W_{n},σ^{n}} W_{n}[u])^{P D}. Its inverse map (W_{n}(A′_{∞}) ⊗_{W_{n},σ^{n}} W_{n}[u])^{P D} → H_{crys}^{0}(𝔄′_{∞}/p^{n}/S_{n}) is defined as follows. For any affine thickening U → T of 𝔄′_{∞}/p^{n} over S_{n}, we define a map (W_{n}(A′_{∞}) ⊗_{W_{n},σ^{n}} W_{n}[u])^{P D} → Γ(U, O_{U}) by (x_{0}, …, x_{n−1}) ⊗ u ↦ u ∑_{k=0}^{n−1} p^{k}t̂_{k}^{p^{n−k}}, where t̂_{k} is a lifting of x_{k} in Γ(T, O_{T}). This is a welldefined ring homomorphism, patches in a non-affine case and induces the inverse map of the natural projection.

Let us consider a surjection W_{n}[u][X′_{0}^{p^{−∞}}, …, X′_{r}^{p^{−∞}}] → 𝔄′_{∞}/p^{n} = W_{n}[X′_{0}^{p^{−∞}}, …, X′_{r}^{p^{−∞}}]/(E(X′_{0}^{p}), f_{1}, …, f_{s}) which sends u to X′_{0}^{p} and X′_{k}^{p^{−i}} to its image for any k, i. Let us denote its kernel (u − X′_{0}^{p}, E(X′_{0}^{p}), f_{1}, …, f_{s}) by I. Taking its divided power envelope with respect to I and compatibility with the natural divided power structure on pW, we get a surjection (W_{n}[u][X′_{0}^{p^{−∞}}, …, X′_{r}^{p^{−∞}}])^{P D} → 𝔄′_{∞}/p^{n}. This map is S-linear, where 𝔄′_{∞}/p^{n} is considered as an Salgebra by u ↦ X′_{0}^{p}. Thus this surjection defines a thickening of 𝔄′_{∞}/p^{n} over S_{n} and we get the natural projection (W_{n}(A′_{∞})⊗_{W_{n},σ^{n}} W_{n}[u])^{P D} → (W_{n}[u][X′_{0}^{p^{−∞}}, …, X′_{r}^{p^{−∞}}])^{P D}.

Conversely, a surjection W_{n}[u][X′_{0}^{p^{−∞}}, …, X′_{r}^{p^{−∞}}] → W_{n}(A′_{∞})⊗_{W_{n},σ^{n}} W_{n}[u] sending u to 1 ⊗ u and X′_{k}^{p^{−i}} to [X′_{k}^{p^{−i−n}}] makes the following diagram commutative;

W_{n}[u][X′_{0}^{p^{−∞}}, …, X′_{r}^{p^{−∞}}] ⟶ W_{n}(A′_{∞}) ⊗_{W_{n},σ^{n}} W_{n}[u]
↓  ↓
𝔄′_{∞}/p^{n}  𝔄′_{∞}/p^{n}.

Therefore this surjection induces (W_{n}[u][X′_{0}^{p^{−∞}}, …, X′_{r}^{p^{−∞}}])^{P D} → (W_{n}(A′_{∞}) ⊗_{W_{n},σ^{n}} W_{n}[u])^{P D}. We see that this map is the inverse to the natural projection by the definition. Thus we get an identification O_{n,π}(𝔄′_{∞}) = (W_{n}[u][X′_{0}^{p^{−∞}}, …, X′_{r}^{p^{−∞}}])^{P D} respecting the filtration and the Frobenius. Then [5, Lemme 2.3.2] and [4, 3.20, Remark 8] conclude the proof.

□

We insert here the next lemma for the sake of references.

Lemma 2.4. Let ψ_{1}, …, ψ_{s} ∈ k[X′_{0}^{p^{−∞}}, …, X′_{r}^{p^{−∞}}] satisfying ψ_{k}^{p} = f̄_{k}. Then O_{1,π}(𝔄′_{∞}) is isomorphic up to a σ-twist to

⨁_{m_{0},...,m_{s+1}∈ℤ_{≥0}}A′_{∞}[u − X′_{0}]/(u − X′_{0})^{p}γ_{pm_{0}}(X′_{0}^{e})γ_{pm_{1}}(ψ_{1}) ⋯

γ_{pm_{s}}(ψ_{s})γ_{pm_{s+1}}(u − X′_{0}).

Proof. The sequence u − X′_{0}^{p}, X′_{0}^{ep}, f̄_{1}, …, f̄_{s} is a regular in k[u][X′_{0}^{p^{−∞}}, …, X′_{r}^{p^{−∞}}]. Their inverse images in (A′_{∞} ⊗_{k,σ} k[u])^{P D} are u − X′_{0}, X′_{0}^{e}, ψ_{1}, …, ψ_{s}, respectively. Thus the assertion follows from the proof of [10, Proposition 1.7].

□

From Lemma 2.3, we have a diagram

J_{n+1,π}|_{O_{K_{1}}} ⟶ J_{n,π}|_{O_{K_{1}}} ⟶^{ϕ_{1}} O_{n,π}|_{O_{K_{1}}} ⟶^{×p} O_{n+1,π}|_{O_{K_{1}}}
↓  ↓  ↓  ↓
J_{n+1,π_{1}} ⟶ J_{n,π_{1}}  ⟶^{ϕ_{1}} O_{n,π_{1}}  ⟶^{×p} O_{n+1,π_{1}},

where the vertical arrows are the pull-backs and the left and right squares are commutative. The compositions of the horizontal maps are ϕ. Thus we see that the middle square is also commutative. In other words, the map pr_{𝔛′}^{∗} is compatible with ϕ_{1}. Therefore, we get the morphism of (Ab/O_{K_{1}})

Ψ_{M} : Gr(M)|_{O_{K_{1}}} → Gr(M′).

Theorem 2.5. The canonical map Ψ_{M} is an isomorphism.

Proof. The sheaves of both sides come from finite flat group schemes Gr(M)×_{O_{K}} O_{K_{1}} and Gr(M′). Thus the bijectivity can be checked after taking the functor Mod. In other words, it suffices to show that

Φ_{M} : M ⊗_{S} S′ → Hom_{(Ab/O_{K_{1}})}(Hom_{(Ab/O_{K_{1}}),′M}^{S}(M, O_{1,π}|_{O_{K_{1}}}), O_{1,π_{1}}),

defined by m ⊗ s′ ↦ (f ↦ s′.pr^{∗}(f(m))) is an isomorphism of ′M′. Here we denote by pr^{∗} the pull-back map O_{1,π}|_{O_{K_{1}}} → O_{1,π_{1}}. We want by devissage to reduce this to the p-torsion case.

Lemma 2.6. Ext_{′M/S_{1}}^{1}(M, O_{1,π}|_{O_{K_{1}}}) = 0 for any M ∈ M which is killed by p.

Proof. Take some Spf(𝔄) ∈ Spf(O_{K_{1}})_{syn} and an extension

0 → O_{1,π}(𝔄′_{∞}) → E → M → 0.

We have to show that syntomic locally a splitting of E exists. Let {e_{1}, …, e_{d}} be an adapted basis of M ([5, Proposition 2.1.2.5]) and ê_{1}, …, ê_{d} be their lifts to E. We mimic [5, Proposition 4.1.3] and seek for a splitting e_{i} ↦ ê_{i} by modifying ê_{i}’s.

Firstly, we modify ê_{i}’s to respect the filtration. Let r_{j} be the minimal natural number satisfying u^{r_{j}}e_{j} ∈ Fil^{1}M. There exists δ_{j} ∈ O_{1,π}(𝔄′_{∞}) such that u^{r_{j}} ê_{j} + δ_{j} ∈ Fil^{1}E. By Lemma 2.4, we can decompose δ_{j} as δ_{j} = δ_{j,0} + δ_{j,1}, where δ_{j,0} ∈ A′_{∞} and δ_{j,1} ∈ J_{1,π}(𝔄′_{∞}). We have u^{e}ê_{j} + u^{e−r_{j}}δ_{j} ∈ Fil^{1}E and u^{e−r_{j}}δ_{j,0} ∈ J_{1,π}(𝔄′_{∞}). As J_{1,π}(𝔄′_{∞}) contains u − X′_{0}, we get X′_{0}^{e−r_{j}}δ_{j,0} = 0, and in particular X′_{0}^{p(e−r_{j})}δ_{j,0}^{p} = 0 in A′_{∞}. Take an lift δ̂_{j,0} of δ_{j,0} in 𝔄′_{∞}, where X′_{0} = π_{1} holds. Then we have π^{e−r_{j}} δ̂_{j,0}^{p} = π^{e}x_{j} for some x_{j} ∈ 𝔄′_{∞}. The ring 𝔄′_{∞} is π-torsion free and we have δ̂_{j,0}^{p} = π^{r_{j}}x_{j}. As 𝔄′_{∞} is perfect, we can take y_{j} ∈ 𝔄′_{∞} satisfying y_{j}^{p} = x_{j}. Then (δ_{j,0} − X′_{0}^{r_{j}}y_{j})^{p} = 0 in A′_{∞}. By the definition of the divided power structure on (k[u][X′_{0}^{p^{−∞}}, …, X′_{r}^{p^{−∞}}])^{P D}, we see that δ_{j,0} − X′_{0}^{r_{j}}y_{j} ∈ J_{1,π}(𝔄′_{∞}) and also δ_{j,0} − u^{r_{j}}y_{j} ∈ J_{1,π}(𝔄′_{∞}).

Now we replace ê_{j} by ê_{j} + y_{j}. Then, u^{r_{j}}(ê_{j} + y_{j}) ≡ −δ_{j} + δ_{j,0} ≡ 0 mod Fil^{1}E. Thus the map e_{j} ↦ ê_{j} + y_{j} respects the filtration.

Next we modify ê_{j} to respect ϕ_{1}. Set (c_{1} ; ⋮ ; c_{d}) = ϕ_{1} (u^{r_{1}} ê_{1} ; ⋮ ; u^{r_{d}} ê_{d})−G (ê_{1} ; ⋮ ; ê_{d}),

where G ∈ GL_{d}(S_{1}) satisfying ϕ_{1} (u^{r_{1}}e_{1} ; ⋮ ; u^{r_{d}}e_{d}) = G (e_{1} ; ⋮ ; e_{d}). We have to find

(δ_{1} ; ⋮ ; δ_{d}) ∈ O_{1,π}(𝔄′_{∞})^{⊕d} such that ϕ_{1} (u^{r_{1}}(ê_{1} + δ_{1}) ; ⋮ ; u^{r_{d}}(ê_{d} + δ_{d})) = G (ê_{1} + δ_{1} ; ⋮ ; ê_{d} + δ_{d}), or

ϕ_{1} (u^{r_{1}}δ_{1} ; ⋮ ; u^{r_{d}}δ_{d}) = G (δ_{1} ; ⋮ ; δ_{d}) − (c_{1} ; ⋮ ; c_{d}). Decompose c_{j} = c_{j,0} + c_{j,1} + c_{j,2},

where c_{j,0} ∈ A′_{∞}, c_{j,1} ∈ (u − X′_{0})A′_{∞} and c_{j,2} ∈ J_{1,π}^{[2]}(𝔄′_{∞}). By linearity,

it suffices to find the solution for ϕ_{1} (u^{r_{1}}δ_{1,k} ; ⋮ ; u^{r_{d}}δ_{d,k}) = G (δ_{1,k} ; ⋮ ; δ_{d,k}) − (c_{1,k} ; ⋮ ; c_{d,k})

for k = 0, 1, 2. We can resolve these equations, taking an appropriate syntomic cover of 𝔄′_{∞} if necessary, just as the proof of [5, Proposition 4.1.3], if we replace Y_{0} and X_{0} there by X′_{0} and X′_{0}^{p}, respectively.

□

Lemma 2.7. Ext_{′M/S_{1}}^{1}(M, O_{∞,π}|_{O_{K_{1}}}) = 0 for any M ∈ M.

Proof. By the Lemma 2.3, the same reasoning as the proof of [5, Lemme 4.1.2] works also in our case and shows that the lemma holds for any M killed by p. Then the definition of the category M and devissage conclude the proof.

□

Now consider an exact sequence in M

0 → M_{1} → M_{2} → M_{3} → 0.

From Lemma 2.7, we get an exact sequence

0 → Hom_{(Ab/O_{K_{1}})}^{′M/S}(M_{3}, O_{∞,π}|_{O_{K_{1}}})

→ Hom_{(Ab/O_{K_{1}})}^{′M/S}(M_{2}, O_{∞,π}|_{O_{K_{1}}}) → Hom_{(Ab/O_{K_{1}})}^{′M/S}(M_{1}, O_{∞,π}|_{O_{K_{1}}}) → 0.

Here we know that Hom_{(Ab/O_{K_{1}})}^{′M/S}(M_{i}, O_{∞,π}|_{O_{K_{1}}}) = Gr(M_{i})|_{O_{K_{1}}}. Thus, from [5, Proposition 4.2.1.5], we have the following commutative diagram whose vertical sequences are exact;

0  0
↓  ↓
M_{1} ⊗_{S} S′ ⟶^{Φ_{M_{1}}} Hom_{(Ab/O_{K_{1}})}(Hom_{(Ab/O_{K_{1}})}^{′M/S}(M_{1}, O_{∞,π}|_{O_{K_{1}}}), O_{∞,π_{1}})
↓  ↓
M_{2} ⊗_{S} S′ ⟶^{Φ_{M_{2}}} Hom_{(Ab/O_{K_{1}})}(Hom_{(Ab/O_{K_{1}})}^{′M/S}(M_{2}, O_{∞,π}|_{O_{K_{1}}}), O_{∞,π_{1}})
↓  ↓
M_{3} ⊗_{S} S′ ⟶^{Φ_{M_{3}}} Hom_{(Ab/O_{K_{1}})}(Hom_{(Ab/O_{K_{1}})}^{′M/S}(M_{3}, O_{∞,π}|_{O_{K_{1}}}), O_{∞,π_{1}})
↓  ↓
0  0

Thus, by devissage, to prove the theorem, we may assume that pM = 0. We have rank_{S′_{1}}(M ⊗_{S} S′) = rank_{S_{1}}(M) and

rank_{S′_{1}}(Hom_{(Ab/O_{K_{1}})}(Hom_{(Ab/O_{K_{1}})}^{′M/S}(M_{1}, O_{∞,π}|_{O_{K_{1}}}), O_{∞,π_{1}}))
= rank_{S′_{1}}(Mod_{K_{1}}(Gr_{K}(M) ×_{O_{K}} O_{K_{1}})) = rank_{S_{1}}(M).

By [5, Lemme 3.3.2], it suffices to show Ker(Φ_{M}) ⊆ Fil^{p}S′_{1}M′.

Take an adapted basis {e_{1}, …, e_{d}} as in the proof of Lemma 2.6. Let m = ∑_{i=1}^{d} s′_{i}e_{i} be an element of Ker(Φ_{M}). Consider the affine algebra R_{M} of Gr_{K}(M) and the element f ∈ Hom_{S}^{′M}(M, O_{1,π}(R_{M})) ≃ Gr_{K}(M)(R_{M}) corresponding to id_{R_{M}}. Then f(e_{i}) ≡ X̄_{i,0} + uX̄_{i,1} + ⋯ + u^{p−1}X̄_{i,p−1} mod J_{1,π}^{[p]}(R_{M}), where X_{i,0}, …, X_{i,p−1} are the canonical generators of R_{M} and X̄_{i,k} its image in R_{M}/p. Here we regard X̄_{i,k} as an element of O_{1,π}(R_{M}) by the natural map R_{M}/p⊗_{k,σ}k[u] → O_{1,π}(R_{M}) (see the proof of [5, Proposition 3.1.1, Proposition 3.1.5]). Let us write f_{1} for the image of f by the natural map Hom_{S}^{′M}(M, O_{1,π}(R_{M})) → Hom_{S}^{′M}(M, O_{1,π}(R′_{M})), where R′_{M} = R_{M} ⊗_{O_{K}} O_{K_{1}}. As m ∈ Ker(Φ_{M}), we have ∑ s′_{i}pr_{R′_{M}}^{∗} (f_{1}(e_{i})) = 0.

Let X̄′_{i,k} be the image of X̄_{i,k} by the natural map R′_{M}/p ⊗_{k,σ} k[v] → O_{1,π_{1}}(R′_{M}). Now we claim that pr_{R′_{M}}^{∗} (X̄_{i,k}) = X̄′_{i,k}. It is sufficient to show this coincidence on an appropriate syntomic cover of R′_{M}. Thus we may consider pr_{R′_{M,∞}}^{∗} : O_{1,π}(R′_{M,∞}) → O_{1,π_{1}}(R′_{M,∞}), where R′_{M,∞} is the perfection of R′_{M} as before. Then the composition

(R′_{M,∞}/p⊗_{k,σ}k[u])^{P D} →^{pr^{∗}} H_{crys}^{0}(R′_{M,∞}/p/S′_{1}) →^{projection} (R′_{M,∞}/p⊗_{k,σ}k[v])^{P D}

maps 1 ⊗ u to 1 ⊗ v^{p} and r ⊗ 1 to r̂^{p} ⊗ 1, where r̂ is a lifting of r by the canonical surjection (R′_{M,∞}/p ⊗_{k,ϕ} k[v])^{P D} → R′_{M,∞}/p. We may take r̂ to be r^{1/p} ⊗ 1. Thus the claim follows.

Therefore we have an equation ∑_{i=1}^{d} s′_{i}(X̄′_{i,0}+v^{p}X̄′_{i,1}+⋯+v^{p(p−1)}X̄′_{i,p−1}) = 0 in O_{1,π_{1}}(R′_{M})/J_{1,π_{1}}^{[p]}(R′_{M}). This equation also holds in O_{1,π_{1}}(R′_{M,∞})/J_{1,π_{1}}^{[p]}(R′_{M,∞}), and its subring R′_{M,∞}/p[v]/(v^{p} − X′_{0}) = R′_{M,∞}/p[v]/(v^{p} − π_{1}) (see [5, Lemme 2.3.2]). As R′_{M,∞} is the direct limit of syntomic covers of R′_{M}, R′_{M}/p is a subring of R′_{M,∞}/p. Thus the equation ∑_{i=1}^{d} s′_{i}(X̄′_{i,0} + v^{p}X̄′_{i,1} + ⋯ + v^{p(p−1)}X̄′_{i,p−1}) = 0 holds in R′_{M}/p[v]/(v^{p} − π_{1}). Let us denote s′_{i} mod v ∈ k by s̄′_{i}. Taking mod v, we have ∑_{i=1}^{d} s̄′_{i}X̄′_{i,0} = 0 in R′_{M}/p[v]/(v, v^{p} − π_{1}) = R′_{M}/π_{1} = R_{M}/π. From the proof of [5, Proposition 3.1.1], we know that X_{1,0}, …, X_{d,0} are linearly independent over k in R_{M}/π. Thus s̄′_{i} = 0 and s′_{i} ∈ vS′_{1}+Fil^{p}S′_{1} for all i. Take s′_{i}^{(1)} ∈ S′_{1} satisfying s′_{i} − vs′_{i}^{(1)} ∈ Fil^{p}S′_{1}. Then we have v ∑_{i=1}^{d} s′_{i}^{(1)}(X̄′_{i,0} +v^{p}X̄′_{i,1} +⋯+v^{p(p−1)}X̄′_{i,p−1}) = 0 in R′_{M}/p[v]/(v^{p} −π_{1}). However, R′_{M}/p ≃ (O_{K_{1}}/p)^{⊕N} ≃ (k[T]/(T^{ep}))^{⊕N} for some N and k[T]/(T^{ep})[v]/(v^{p} − T) ≃ k[v]/(v^{ep^{2}}). Thus R′_{M}/p[v]/(v^{p} − π_{1}) is finite flat over k[v]/(v^{ep^{2}}), and we have ∑_{i=1}^{d} s′_{i}^{(1)}(X̄′_{i,0} + v^{p}X̄′_{i,1} + ⋯ + v^{p(p−1)}X̄′_{i,p−1}) ∈ v^{ep^{2}−1}(R′_{M}/p[v]/(v^{p} − π_{1})). Taking mod v and repeating this procedure shows s′_{i} ∈ v^{ep^{2}}S′_{1} + Fil^{p}S′_{1} = Fil^{p}S′_{1}. In other words, m ∈ Fil^{p}S′_{1}M′. This concludes the theorem.

□

Remark 2.8. In general, let L be a totally ramified extension over K of degree e′ whose uniformizer we denote by π_{L}. When we define S_{L} = S_{π_{L}} as above, there exists a morphism S → S_{L} respecting the filtration and ϕ_{1} if and only if π_{L}^{e′} = πζ_{p−1}^{i} for some i.

3. Rank one calculation

In this section, we calculate the conductor of a Raynaud 𝔽-vector space scheme over O_{K}. The point is that, as we can see from the bound of the conductor ([11, Theorem 7]), it is enough to consider the j-th tubular neighborhood only for j ≤ pe/(p − 1) + ε with sufficiently small ε > 0. For such j, we can compute the tubular neighborhood easily by Lemma 3.4 below.

Let K be a complete discrete valuation field of mixed characteristic (0, p). We write π = π_{K} for its uniformizer and e for its absolute ramification index. We normalize a valuation v_{K} of K as v_{K}(π) = 1 and extend it to the algebraic closure K̄ of K. For a ∈ K̄ and j ∈ ℝ, let D(a, j) denote the closed disc {z ∈ OK̄ | v_{K}(z −a) ≥ j}. This is the underlying subset of a K(a)-affinoid subdomain of the unit disc over K(a).

For integers 0 ≤ s_{1}, …, s_{r} ≤ e, let G(s_{1}, …, s_{r}) denote the Raynaud 𝔽_{p^{r}}-vector space scheme over O_{K} defined by the r equations T_{1}^{p} = π^{s_{1}}T_{2}, T_{2}^{p} = π^{s_{1}}T_{3}, …, T_{r}^{p} = π^{s_{r}}T_{1} ([13]). We set j_{k} = (ps_{k} + p^{2}s_{k−1} + … + p^{k}s_{1} + p^{k+1}s_{r} + p^{k+2}s_{r−1} + ⋯ + p^{r}s_{k+1})/(p^{r} − 1). Before the calculation of c(G(s_{1}, …, s_{r})), we gather some elementary lemmas.

Lemma 3.1. Let a ∈ O_{K} and s = v_{K}(a). Then the affinoid variety X^{j}(K̄) = {x ∈ OK̄ | v_{K}(x^{p} − a) ≥ j} is equal to

\{ \begin{matrix}D(a^{1/p}, j/p) & if j ≤ s + pe/(p − 1), \\ ∐_{i=0}^{p−1} D(a^{1/p}ζ_{p}^{i}, j − e − (p − 1)s/p) & if j > s + pe/(p − 1). \end{matrix}

Proof. We have v_{K}(x^{p} − a) = ∑_{i} v_{K}(x − a^{1/p}ζ_{p}^{i}). If v_{K}(x − a^{1/p}ζ_{p}^{i}) ≥ v_{K}(x − a^{1/p}ζ_{p}^{i′}) for any i′ ≠ i, then v_{K}(x − a^{1/p}ζ_{p}^{i′}) ≤ v_{K}(a^{1/p}ζ_{p}^{i′}(1 − ζ_{p}^{i−i′})) = s/p + e/(p − 1). Thus we have v_{K}(x − a^{1/p}ζ_{p}^{i}) ≥ sup(j/p, j − (p − 1)s/p − e) and

X^{j}(K̄) ⊆ ⋃_{i} D(a^{1/p}ζ_{p}^{i}, sup(j/p, j − (p − 1)s/p − e)).

Suppose that j/p ≥ j−(p−1)s/p−e. Then we have v_{K}(a^{1/p}(1−ζ_{p}^{i})) = s/p + e/(p − 1) ≥ j/p, D(a^{1/p}, j/p) = D(a^{1/p}ζ_{p}^{i}, j/p) for any i and thus

X^{j}(K̄) = D(a^{1/p}, j/p).

When j/p < j−(p−1)s/p−e, we have v_{K}(a^{1/p}(1−ζ_{p}^{i})) = s/p+e/(p− 1) < j−(p−1)s/p−e. This means that if w ∈ D(a^{1/p}ζ_{p}^{i}, j−(p−1)s/p−e) for some i, then v_{K}(w − a^{1/p}ζ_{p}^{i′}) < j − (p − 1)s/p − e for any other i′. Thus the discs D(a^{1/p}ζ_{p}^{i}, j − (p − 1)s/p − e) are disjoint and

X^{j}(K̄) = ∐_{i} D(a^{1/p}ζ_{p}^{i}, j − (p − 1)s/p − e).

These are equalities of the underlying sets of affinoid subdomains of the unit disc over K(a^{1/p}, ζ_{p}). By the universality of an affinoid subdomain, this extends to an isomorphism of affinoid varieties.

□

We can prove the following lemma just in the same way.

Lemma 3.2. The affinoid variety {x ∈ OK̄ | v_{K}(x^{p^{r}} − ax) ≥ j} is equal to

\{ \begin{matrix}D(0, j/p^{r}) & if j ≤ p^{r}v(a)/(p^{r} − 1), \\ ∐_{i=0}^{p^{r}−1} D(σ_{i}, j − v(a)) & if j > p^{r}v(a)/(p^{r} − 1), \end{matrix}

where σ_{i}’s are the roots of X^{p^{r}} = aX.

Lemma 3.3. For g_{1}(Y_{1}, …, Y_{d}), g_{2}(Y_{1}, …, Y_{d}) ∈ K[Y_{1}, …, Y_{d}] and

j_{1} ≥ j_{2}, the affinoid variety {(x, y_{1}, …, y_{d}) ∈ OK̄ × Od ; K̄ | v_{K}(x −

g_{1}(y_{1}, …, y_{d})) ≥ j_{1}, v_{K}(x−g_{2}(y_{1}, …, y_{d})) ≥ j_{2}} is equal to {(x, y_{1}, …, y_{d}) ∈

OK̄×Od ; K̄ | v_{K}(x−g_{1}(y_{1}, …, y_{d})) ≥ j_{1}, v_{K}(g_{1}(y_{1}, …, y_{d})−g_{2}(y_{1}, …, y_{d})) ≥

j_{2}}.

Proof. For fixed (x, y), these two conditions are equivalent. The universality of an affinoid subdomain proves the lemma.

□

Lemma 3.4. Let a ∈ O_{K} and s = v_{K}(a). If j ≤ pe/(p − 1) + s, then the affinoid variety X^{j}(K̄) = {(x, y) ∈ OK̄ × OK̄ | v_{K}(x^{p} − ay^{p^{n}}) ≥ j} is equal to {(x, y) ∈ OK̄ × OK̄ | v_{K}(x − a^{1/p}y^{p^{n−1}}) ≥ j/p}.

Proof. Lemma 3.1 shows that the fiber of the second projection X^{j}(K̄) → OK̄ at y is equal to

\{ \begin{matrix}D(a^{1/p}y^{p^{n−1}}, j/p) if j ≤ s + p^{n−1}v_{K}(y) + pe/(p − 1), \\ ∐_{i=0}^{p−1} D(a^{1/p}ζ_{p}^{i}y^{p^{n−1}}, j − e − (p − 1)(s + p^{n−1}v_{K}(y))/p) otherwise. \end{matrix}

Thus we have X^{j}(K̄) = {(x, y) ∈ OK̄ × OK̄ | v_{K}(x − a^{1/p}y^{p^{n−1}}) ≥ j/p} for j ≤ pe/(p − 1) + s. This is the underlying set of a K(a^{1/p})-affinoid variety. Again this equality extends to an isomorphism over K(a^{1/p}).

□

Now we proceed to the proof of the main theorem of this section.

Theorem 3.5. c(G(s_{1}, …, s_{r})) = sup_{k} j_{k}.

Proof. We may assume that j_{r} is the supremum of j_{k}’s. If j_{r} = 0, then G(s_{1}, …, s_{r}) is etale and c(G(s_{1}, …, s_{r})) = 0. Thus we may assume j_{r} > 0. Consider the homomorphism of O_{K}-algebras

A = O_{K}[T_{1}, …, T_{r}]/(T_{1}^{p} − π^{s_{1}}T_{2}, …, T_{r}^{p} − π^{s_{r}}T_{1}) →

B = O_{K}[W, T_{2}, …, T_{r}]/(W^{p^{r}} − π^{s_{1}}T_{2}, T_{2}^{p} − π^{s_{2}}T_{3}, …,
T_{r−1}^{p} − π^{s_{r−1}}T_{r}, T_{r}^{p} − π^{s_{r}}W^{p^{r−1}}),

defined by T_{1} ↦ W^{p^{r−1}}. This induces a surjection of K-affinoid varieties

X_{B}^{j}(K̄) ∋ (w, t_{2}, …, t_{r}) ↦ (w^{p^{r−1}}, t_{2}, …, t_{r}) ∈ X_{A}^{j}(K̄),

where

X_{A}^{j}(K̄) = {(t_{1}, …, t_{r}) ∈ Or ; K̄ | v_{K}(t_{1}^{p} − π^{s_{1}}t_{2}) ≥ j, …,

v_{K}(t_{r−1}^{p} − π^{s_{r−1}}t_{r}) ≥ j, v_{K}(t_{r}^{p} − π^{s_{r}}t_{1}) ≥ j} and

X_{B}^{j}(K̄) = {(w, t_{2}, …, t_{r}) ∈ Or ; K̄ | v_{K}(w^{p^{r}} − π^{s_{1}}t_{2}) ≥ j,

v_{K}(t_{2}^{p} − π^{s_{2}}t_{3}) ≥ j, …, v_{K}(t_{r}^{p} − π^{s_{r}}w^{p^{r−1}}) ≥ j}.

These are affinoid subdomains of the r-dimensional unit polydisc over K. We calculate a jump of {F^{j}(B)}_{j∈ℚ_{>0}} at first.

Lemma 3.6. If j_{r} < pe/(p − 1), then the first jump of {F^{j}(B)}_{j∈ℚ_{>0}} occurs at j = j_{r} and ♯F^{j_{r}}(B) = p^{r}.

Note that the base change from K to a finite extension L multiplies s_{i}’s, j_{i}’s and e by the ramification index of L/K. Thus, to prove Lemma 3.6 and Theorem 3.5, we may assume that p^{r−1} divides s_{i}’s and e.

Proof. Consider the K-affinoid variety X_{B}^{j} for j ≤ pe/(p − 1). Then the iterative use of Lemma 3.4 and Lemma 3.3 shows that the affinoid variety X_{B}^{j}(K̄) is equal to

{v_{K}(w^{p^{r}} − π^{(s_{r}+ps_{r−1}+...+p^{r−1}s_{1})/p^{r−1}}w) ≥ pl_{1}(j), v_{K}(t_{2} − g_{2}(w)) ≥ u_{2}, v_{K}(t_{3} − g_{3}(t_{2}, w)) ≥ u_{3}, …, v_{K}(t_{r} − g_{r}(t_{r−1}, w)) ≥ u_{r}},

where l_{i}(j), g_{i}(t_{i−1}, w), g_{2}(w) and u_{i} are defined as follows; • l_{r}(j) = j/p,

• l_{i−1}(j) = inf(j, l_{i}(j) + s_{i−1})/p,

• g_{i}(t_{i−1}, w) = t_{i−1}^{p}/π^{s_{i−1}} and u_{i} = j − s_{i−1} if j ≥ l_{i}(j) + s_{i−1},

• g_{i}(t_{i−1}, w) = π^{s_{r}+ps_{r−1}+...+p^{r−i}s_{i}/p^{r−i+1}}w^{p^{i−2}} and u_{i} = l_{i}(j) if j <

l_{i}(j) + s_{i−1},
• g_{2}(w) = g_{2}(w^{p^{r−1}}, w).

Note that l_{i}(j) is a strictly monotone increasing function of j. This affinoid variety is isomorphic to the product of the affinoid variety {w ∈ OK̄ | v(w^{p^{r}} −π^{(s_{r}+ps_{r−1}+...+p^{r−1}s_{1})/p^{r−1}}w) ≥ pl_{1}(j)} and discs. Therefore, from Lemma 3.2, we see that the first jump of {F^{j}(B)}_{j∈ℚ_{>0}} occurs at j such that pl_{1}(j) = j_{r}, provided this j satisfies 0 < j < pe/(p − 1). Moreover, then we have ♯F^{j}(B) = p^{r}. Thus the following lemma and the strict monotonicity of l_{1} terminate the proof of Lemma 3.6. □

Lemma 3.7. l_{1}(j_{r}) = j_{r}/p.

Proof. Suppose that there is k such that l_{k}(j_{r}) = j_{r}/p and j_{r} ≥ l_{k′}(j_{r})+ s_{k′} for any 1 < k′ ≤ k. Then we have l_{1}(j_{r}) = inf(j_{r}, (j_{r} + ps_{k−1} + p^{2}s_{k−2} + … + p^{k−1}s_{1})/p^{k−1})/p and the assumption j_{k−1} ≤ j_{r} implies l_{1}(j_{r}) = j_{r}/p.

□

On the other hand, let s = (s_{r} + ps_{r−1} + … + p^{r−1}s_{1})/p^{r−1} and σ_{0}, …, σ_{p^{r}−1} be the roots of the equation X^{p^{r}} − π^{s}X = 0. Then we see that the images by w ↦ w^{p^{r−1}} of the discs D(σ_{i}, pl_{1}(j) − s) are disjoint for j > j_{r}. Hence the surjection π_{0}(X_{B}^{j}(K̄)) → π_{0}(X_{A}^{j}(K̄)) is bijective for 0 < j ≤ pe/(p − 1) and the first (and the last) jump of {F^{j}(A)}_{j∈ℚ_{>0}} also occurs at j_{r}, provided j_{r} < pe/(p − 1).

When j_{r} = pe/(p−1), we see that s_{k} = e > 0 for any k. Thus we can use Lemma 3.4 for j < pe/(p−1)+ε with sufficiently small ε > 0. Then, by the same reasoning as above, we conclude that c(A) = pe/(p − 1).

□

4. Hasse-Arf theorem for 𝔽_{p}-rank two case

Let K be as in section 1. In this section, we prove Conjecture 1.2 in the case where 𝔽 = 𝔽_{p} and G(K̄) is reducible.

Theorem 4.1. Let G be a finite flat group scheme over O_{K} of rank p^{2} which is killed by p and reducible. Then the I_{K}-representation G(K̄) contains θ_{K,l}^{k}, where k/l mod ℤ is the prime-to-p part of the conductor c(G).

We prove this theorem by a lengthy calculation of the conductor. The point is that, on the one hand, to check the assertion on a character, we may restrict to G_{K_{∞}} by the full faithful theorem of Breuil ([6, Theorem 3.4.3]) and on the other hand, we can describe a defining equation of R_{M} explicitly in terms of M after the base change to K_{∞}. By abuse of notation, we may write F^{j}(M) for F^{j}(Gr(M)) and c(M) for c(Gr(M)). We fix once and for all a (p − 1)-st root π_{1}^{1/(p−1)} of π_{1} and set π^{1/(p−1)} = π_{1}^{p/(p−1)}.

Proof. It is sufficient to show the theorem in the case of k = k̄. Let M = Mod_{K}(G) be the filtered ϕ_{1}-module of G. By assumption, we have an exact sequence in ′M

0 → M(s) → M → M(r) → 0

for some integers 0 ≤ r, s ≤ e, where M(s) is the filtered ϕ_{1}-module defined by M(s) = S_{1}e, Fil^{1}M(s) = u^{s}S_{1}e and ϕ_{1}(u^{s}e) = e. We have Gr_{K}(M(s)) ≃ G(e − s) by the notation of Section 3. By [7, Lemma 5.2.2], we may assume that M̃ = M/Fil^{p}SM is of the following type;

• M̃ = S̃_{1}e_{0} ⊕ S̃_{1}e_{1}, where S̃ = S/Fil^{p}S = k[u]/(u^{ep}),

• Fil^{1}M̃ = ⟨u^{s}e_{0}, u^{r}e_{1} + fe_{0}⟩, where f ∈ u^{sup(0,r+s−e)}S̃_{1}
• ϕ_{1}(u^{s}e_{0}) = e_{0} and ϕ_{1}(u^{r}e_{1} + fe_{0}) = e_{1}.

Put m = v_{u}(f). Then we can take an adapted basis of M̃ as follows.

Case A: (e_{0}, s), (e_{1} + (f/u^{r})e_{0}, r) if m ≥ r
Case B: (e_{0}, s), (e_{1}, r) if s ≤ m < r

Case C: ((f/u^{m})e_{0}+u^{r−m}e_{1}, m), ((u^{m}/f)e_{1}, r+s−m) if m < r, s Before starting calculation of the conductor, we show the following lemma.

Lemma 4.2. For any exact sequences of finite flat group schemes over O_{K}

0 → G_{1} → G_{2} → G_{3} → 0,

we have c(G_{2}) ≥ c(G_{1}), c(G_{3}).

Proof. The inequality c(G_{2}) ≥ c(G_{3}) follows from [1, Lemme 2.10]. Let us show c(G_{2}) ≥ c(G_{1}). Taking a sufficiently large base change, we may assume that G_{K} acts trivially on G_{3}(K̄). Let H be the maximal prolongation of G_{3} ([13]) and G′ = G_{2} ×_{G_{3}} H. The group scheme G′ is also finite flat over O_{K}. We know H is constant. Thus we have c(G′) = c(G_{1}). However, the natural map G′ → G_{2} is a prolongation. Therefore c(G′) ≤ c(G_{2}) (see the proof of [11, Theorem 7]).

□

A. m ≥ r. In this case, we have

ϕ_{1} (\begin{matrix}u^{s}e_{0} \\ u^{r}e_{1} + fe_{0} \end{matrix}) = (\begin{matrix}e_{0} \\ e_{1} \end{matrix}) = (\begin{matrix}u^{s}e_{0} \\ (e_{1} + (f/u^{r})e_{0}) − (f/u^{r})e_{0} \end{matrix})

= (\begin{matrix}1 & 0 \\ −(f/u^{r}) & 1 \end{matrix}) (\begin{matrix}e_{0} \\ e_{1} + (f/u^{r})e_{0} \end{matrix}) .

We calculate the conductor c(R′_{M}) = pc(R_{M}). From Theorem 2.5, it is equal to c(Gr_{K_{1}}(M′)) = c(M′). We see that (e_{0}, ps), (e_{1}+(f(v^{p})/v^{pr})e_{0}, pr) is an adapted basis of M̃′ and

ϕ_{1} (\begin{matrix}v^{ps}e_{0} \\ v^{pr}e_{1} + f(v^{p})e_{0} \end{matrix}) = (\begin{matrix}1 & 0 \\ −f(v^{p})/v^{pr} & 1 \end{matrix}) (\begin{matrix}e_{0} \\ e_{1} + (f(v^{p})/v^{pr})e_{0} \end{matrix}) .

Consider the surjection O_{K_{2}} → O_{K_{2}}/p ⊗_{k,ϕ} k ≃ k[v]/(v^{ep^{2}}) where the last map is k-linear and maps π_{1} to u. The matrix above lifts by this

surjection to (\begin{matrix}1 & 0 \\ f_{0}(π_{1}) & 1 \end{matrix}), where f_{0}(v) = −f^{σ^{−1}}(v)/v^{r}. Then, from

[5, Proposition 3.1.2], we see that

R′_{M} = O_{K_{1}}[X_{1}, X_{2}]/(X_{1}^{p}+π^{e−s}F (π)^{−1}X_{1}, X_{2}^{p}+π^{e−r}F (π)^{−1}(X_{2}+f_{0}(π_{1})X_{1})).

Let us calculate the affinoid variety

\begin{matrix}X_{M′}^{j}(K̄) = & \{ (x_{1}, x_{2}) ∈ OK̄_{1} × OK̄_{1} | v_{K_{1}}(x_{1}^{p} + π^{e−s}F (π)^{−1}x_{1}) ≥ j, \\ v_{K_{1}}(x_{2}^{p} + π^{e−r}F (π)^{−1}(x_{2} + f_{0}(π_{1})x_{1})) ≥ j \}. \end{matrix}

Note that the second inequality is equivalent to

v_{K_{1}}(x_{1} + f_{0}(π_{1})^{−1}x_{2} + F (π)(f_{0}(π_{1})π^{e−r})^{−1}x_{2}^{p}) ≥ j − p(e − r) − (m − r).

We have c(M′) ≥ c_{K_{1}}(G(e − s) ×_{O_{K}} O_{K_{1}}) = p^{2}(e − s)/(p − 1) from Lemma 4.2. Thus we may suppose j > p^{2}(e − s)/(p − 1) and that the affinoid variety defined by the first inequality in the definition of X_{M′}^{j}(K̄) splits. Thus we have

\begin{aligned}X_{M′}^{j}(K̄) = ∐_{l=0}^{p−1}\{ & (x_{1}, x_{2}) ∈ OK̄_{1} × OK̄_{1} | v_{K_{1}}(x_{1} − σ_{s,l}) ≥ j − p(e − s), \\ & v_{K_{1}}(x_{2}^{p} + π^{e−r}F (π)^{−1}x_{2} + π^{e−r}F (π)^{−1}f_{0}(π_{1})x_{1}) ≥ j\}, \end{aligned}

where σ_{s,l} = \begin{cases}0 & (l = 0) \\ π^{(e−s)/(p−1)}ζ_{p−1}^{l} & (l = 1, …, p − 1). \end{cases} Let us denote its l-

th component by X_{M′,l}^{j}. We have a surjection of G_{K}-modules F^{j}(M′) → F^{j}(M(s)) ([1, Lemme 2.10]) and the inverse image of σ_{s,l} ∈ F^{j}(M(s)) by this surjection is equal to π_{0}(X_{M′,l}^{j})K̄. Thus X_{M′}^{j} splits if and only if X_{M′,0}^{j} splits.

If s ≥ r, we have v_{K_{1}}(π^{e−r}F (π)^{−1}f_{0}(π_{1})x_{1}) ≥ j and

\begin{matrix}X_{M′,0}^{j}(K̄) = \{ & (x_{1}, x_{2}) ∈ OK̄_{1} × OK̄_{1} | \\ v_{K_{1}}(x_{1}) ≥ j − p(e − s), v_{K_{1}}(x_{2}^{p} + π^{e−r}F (π)^{−1}x_{2}) ≥ j \}. \end{matrix}

Thus c(M′) = sup(p^{2}(e−s)/(p−1), p^{2}(e−r)/(p−1)) = p^{2}(e−r)/(p−1).

If s < r and m ≥ p(r − s) + r, Lemma 3.3 shows that X_{M′,0}^{j}(K̄) is the same as the case above and we have c(M′) = p^{2}(e − s)/(p − 1). Suppose s < r and m < p(r − s) + r. Then we have

X_{M′,0}^{j}(K̄) = {(x_{1}, x_{2}) ∈ OK̄_{1} × OK̄_{1} |
v_{K_{1}}(x_{2}^{p} + π^{e−r}F (π)^{−1}(x_{2} + f_{0}(π_{1})x_{1}) ≥ j,

v_{K_{1}}(x_{2}^{p} + π^{e−r}F (π)^{−1}x_{2}) ≥ j − (p(r − s) − (m − r)) }.

This affinoid variety splits if and only if j > p^{2}(e−r)/(p−1)+p(r−s)− (m − r). The conductor equals sup(p^{2}(e − r)/(p − 1) + p(r − s) − (m − r), p^{2}(e−s)/(p−1)). We see that p^{2}(e−r)/(p−1)+p(r−s)−(m−r) ≥ p^{2}(e − s)/(p − 1) if and only if (ps − r)/(p − 1) ≥ m. This does not occur, since we have s < r ≤ m and (ps − r)/(p − 1) ≤ s < m. Thus we have c(M′) = p^{2}(e − s)/(p − 1).

To terminate the proof of the theorem in Case A, we must show that G(K̄) contains θ_{K,p−1}^{e−s} if c(M) ≡ p(e − s)/(p − 1) mod 1/p^{∞}ℤ. Note that, if p(e − r)/(p − 1) ≡ p(e − s)/(p − 1) mod 1/p^{∞}ℤ, then we have p^{M}(e − r) ≡ p^{M}(e − s) mod (p − 1)ℤ for some integer M and θ_{K,p−1}^{e−r} = θ_{K,p−1}^{e−s}. Thus we may restrict our attention to the case s < r. By virtue of the full faithful theorem of Breuil, it suffices to show that the G_{K_{1}}-module Gr(M′)(K̄) contains θ_{K_{1},p−1}^{e−s}.

We identify the finite G_{K_{1}}-set Gr(M′)(K̄) with the solution (X, Y) of the equation

\{ \begin{matrix}X^{p} + π^{e−r}F (π)^{−1}X = 0 \\ Y^{p} + π^{e−r}F (π)^{−1}Y + π^{e−r}f_{0}(π_{1})F (π)^{−1}X = 0. \end{matrix}

Consider the equation Y^{p} + π^{e−r}F (π)^{−1}Y + π^{e−r}f_{0}(π_{1})F (π)^{−1}σ_{s,l} = 0. Its Newton polygon has an internal vertex if and only if (p − 1)/p(m − r + p(e − r) + p(e − s)/(p − 1)) − p(e − r) > 0, which is equivalent to r > s. Thus there is one and only one root Y_{l} of this equation which satisfies v_{K_{1}}(Y_{l}) = m − r + p(e − s)/(p − 1) for

each l > 0. Define α ∈ O× ; K̄ by Y_{l} = α^{−1}π_{1}^{m−r+p(e−s)/(p−1)}. Then

α is the root of the equation T^{p−1} + aT^{p−2} + b = 0, where a, b ∈ O_{K_{1}} with v_{K_{1}}(a) = 0 and v_{K_{1}}(b) > 0. Hensel’s lemma shows that α ∈ K_{1}^{nr}. Thus we have g(Y_{l}) = Y_{l}g(π_{1}^{p(e−s)/(p−1)})/π_{1}^{p(e−s)/(p−1)} for any g ∈ I_{K_{1}}. Denote by P (resp. Q) an element (X, Y) = (0, σ_{r,1}) (resp. (X, Y) = (σ_{s,1}, Y_{1})) of V = Gr(M′)(K̄). From [5, Lemme 3.1.7], we see that the subspace G(p(e − r))(K̄) ⊆ V can be identified with a subset {(0, 0), (0, σ_{r,1}), …, (0, σ_{r,p−1})}. Thus P and Q form a basis of V. For the Galois extension L = K_{1}(π_{1}^{1/(p−1)}) of degree p − 1 over K_{1}, we see that I_{L} acts trivially on P and Q. This shows that the image of I_{K_{1}} → Aut(V) has an order prime to p. Therefore we have V = θ_{K_{1},p−1}^{e−s} ⊕ θ_{K_{1},p−1}^{e−r} as an I_{K_{1}}-module. This concludes the proof in the case A.

B. s ≤ m < r. We have

ϕ_{1} (\begin{matrix}u^{s}e_{0} \\ u^{r}e_{1} \end{matrix}) = ϕ_{1} (\begin{matrix}u^{s}e_{0} \\ (u^{r}e_{1} + fe_{0}) − (f/u^{s})u^{s}e_{0} \end{matrix})

= (\begin{matrix}e_{0} \\ e_{1} − (f^{σ}(u^{p})/u^{ps})e_{0} \end{matrix}) = (\begin{matrix}1 & 0 \\ −f^{σ}(u^{p})/u^{ps} & 1 \end{matrix}) (\begin{matrix}e_{0} \\ e_{1} \end{matrix}) .

Set g_{0}(u) = −f(u)/u^{s}. Then a matrix (\begin{matrix}1 & 0 \\ g_{0}(π) & 1 \end{matrix}) maps to the matrix

above by the surjection O_{K_{1}} → O_{K_{1}} ⊗_{k,σ} k ≃ k[u]/(u^{ep}). Then we see that

R_{M} = O_{K}[X_{1}, X_{2}]/(X_{1}^{p}+π^{e−s}F (π)^{−1}X_{1}, X_{2}^{p}+π^{e−r}F (π)^{−1}(X_{2}+g_{0}(π)X_{1})).

An affinoid variety X_{M}^{j} we must calculate is

\begin{aligned}\{ & (x_{1}, x_{2}) ∈ OK̄ × OK̄ | v_{K}(x_{1}^{p} + π^{e−s}F (π)^{−1}x_{1}) ≥ j, \\ & v_{K}(x_{1} + g_{0}(π)^{−1}x_{2} + F (π)(g_{0}(π)π^{e−r})^{−1}x_{2}^{p}) ≥ j − (e − r) − (m − s) \} \end{aligned}

Again it is sufficient to suppose j > p(e − s)/(p − 1) and consider an affinoid variety

\begin{aligned}\{ & (x_{1}, x_{2}) ∈ OK̄ × OK̄ | v_{K}(x_{1}) ≥ j − (e − s), \\ & v_{K}(x_{1} + g_{0}(π)^{−1}x_{2} + F (π)(g_{0}(π)π^{e−r})^{−1}x_{2}^{p}) ≥ j − (e − r) − (m − s) \}. \end{aligned}

By the assumption m < r and Lemma 3.3, this is equal to an affinoid variety

\begin{aligned}\{ & (x_{1}, x_{2}) ∈ OK̄ × OK̄ | v_{K}(x_{2}^{p} + π^{e−r}F (π)^{−1}x_{2}) ≥ j − (r − m), \\ & v_{K}(x_{1} + g_{0}(π)^{−1}x_{2} + F (π)(g_{0}(π)π^{e−r})^{−1}x_{2}^{p}) ≥ j − (e − r) − (m − s) \}, \end{aligned}

which splits if and only if j > r − m + p(e − r)/(p − 1). Therefore we get c(M) = r − m + p(e − r)/(p − 1) if m ≤ (ps − r)/(p − 1) and p(e − s)/(p − 1) if (ps − r)/(p − 1) < m < r. In the latter case, the verbatim arguments as in Case A shows that G(K̄) = θ_{K,p−1}^{e−s} ⊕ θ_{K,p−1}^{e−r} as an I_{K}-module.

C. r, s > m. In this case,

ϕ_{1} (\begin{matrix}u^{r}e_{1} + fe_{0} \\ (u^{r+s}/f)e_{1} \end{matrix}) = ϕ_{1} (\begin{matrix}u^{r}e_{1} + fe_{0} \\ (u^{s}/f)(u^{r}e_{1} + fe_{0}) − u^{s}e_{0} \end{matrix})

= (\begin{matrix}e_{1} \\ (u^{ps}/f^{σ}(u^{p}))e_{1} − e_{0} \end{matrix})

= (\begin{matrix}0 & f/u^{m} \\ −u^{m}/f & (f/u^{m})(u^{r}/f + (u^{s}/f)^{p}) \end{matrix}) (\begin{matrix}(f/u^{m})e_{0} + u^{r−m}e_{1} \\ (u^{m}/f)e_{1} \end{matrix}) .

Again we consider M′ = M ⊗_{S} S′. Then the last matrix is equal to

(\begin{matrix}0 & f(v^{p})/v^{pm} \\ −v^{pm}/f(v^{p}) & (f(v^{p})/v^{pm})(v^{pr}/f(v^{p}) + (v^{ps}/f(v^{p}))^{p}) \end{matrix}). We can take

as a lifting of this matrix (\begin{matrix}0 & c \\ −1/c & c(π_{1}^{r−m}/c + (π_{1}^{s−m}/c)^{p}) \end{matrix}) ∈ GL_{2}(O_{K_{1}})

with c = f^{σ^{−1}}(π_{1})/π_{1}^{m}. Thus we get

\begin{matrix}R′_{M} = O_{K_{1}}[X_{1}, X_{2}]/(X_{1}^{p}+ & cπ^{e−m}F (π)^{−1}X_{2}, \\ X_{2}^{p} + π^{e+m−(r+s)}F (π)^{−1}(−c^{−1}X_{1} + dX_{2})), \end{matrix}

where d = c(π_{1}^{r−m}/c + (π_{1}^{s−m}/c)^{p}). As in Section 3, we firstly calculate the conductor of R̃′_{M} = R′_{M}[W]/(W^{p} − X_{2}).

Consider an affinoid variety

\begin{aligned}Y_{M′}^{j}(K̄) = \{ & (x, w) ∈ OK̄ × OK̄ | v_{K_{1}}(x^{p} + π^{e−m}F (π)^{−1}cw^{p}) ≥ j, \\ & v_{K_{1}}(w^{p^{2}} + π^{e+m−(r+s)}F (π)^{−1}(−c^{−1}x + dw^{p})) ≥ j \}. \end{aligned}

From the bound of the conductor [11, Theorem 7], we may suppose j ≤ p^{2}e/(p − 1) + ε with ε > 0 sufficiently small. Now we have e > m, and Lemma 3.4 shows that the first inequality in the definition of Y_{M′}^{j} is equivalent to v_{K_{1}}(x + π_{1}^{e−m}c^{1/p}F (π)^{−1/p}w) ≥ j/p. On the other hand, the second inequality can be written also as v_{K_{1}}(x − cF (π)π^{r+s−e−m}w^{p^{2}} − cdw^{p}) ≥ j − p(e + m − (r + s)). Put λ_{0} = π_{1}^{(p+1)e+(p−1)m−p(r+s)}/(F (π)^{(p+1)/p}c^{(p−1)/p}) and λ_{1} = dπ_{1}^{p(e+m−(r+s))}/F (π). Using Lemma 3.3, we see that this affinoid variety is equal to

\begin{aligned}Y_{M′}^{j}(K̄) = \{ & (x, w) ∈ OK̄ × OK̄ | v_{K_{1}}(x + π_{1}^{e−m}c^{1/p}F (π)^{−1/p}w) ≥ j/p, \\ & v_{K_{1}}(w^{p^{2}} + λ_{1}w^{p} + λ_{0}w) ≥ j \} \end{aligned}

if j ≤ p^{2}(e + m − (r + s))/(p^{2} − 1) and

Y_{M′}^{j}(K̄) = { (x, w) ∈ OK̄ × OK̄ |
v_{K_{1}}(w^{p^{2}} + π^{e+m−(r+s)}F (π)^{−1}(−c^{−1}x + dw^{p})) ≥ j,

v_{K_{1}}(w^{p^{2}} + λ_{1}w^{p} + λ_{0}w) ≥ j/p + p(e + m − (r + s)) }

if j > p^{2}(e + m − (r + s))/(p^{2} − 1).

Lemma 4.3. Put P (W) = W^{p^{2}} + λ_{1}W^{p} + λ_{0}W. Then an affinoid variety { w ∈ OK̄ | v_{K_{1}}(P (w)) ≥ j } splits if and only if

j > \{ \begin{matrix}p^{2}(e − r)/(p − 1) − (p(s − m) & − (r − m)) \\ if m < (ps − r)/(p − 1) \end{matrix} ; p^{2}(e − s)/(p − 1) − p(r − m) if m ≥ (ps − r)/(p − 1).

Proof. Set μ_{k} = v_{K_{1}}(λ_{k}). Consider the Newton polygon of the polynomial P (W). We have μ_{0}(p^{2} − p)/(p^{2} − 1) − μ_{1} = p(r − m + s − m)/(p + 1) − v_{K_{1}}(d) and

v_{K_{1}}(d) = \{ \begin{matrix}r − m & if m < (ps − r)/(p − 1) \\ r − m + v_{K_{1}}(c + c^{p}) & if m = (ps − r)/(p − 1) \\ p(s − m) & if m > (ps − r)/(p − 1). \end{matrix}

In the first and third case, P (W) has p − 1 roots of valuation (μ_{0} − μ_{1})/(p − 1) and p^{2} − p roots of valuation μ_{1}/(p^{2} − p). Let w be one of these roots and V = W − w. Then P (V + w) = V^{p^{2}} + (_{p^{2}}C_{p}w^{p^{2}−p} + λ_{1})V^{p}+(p^{2}w^{p^{2}−1}+pλ_{1}w^{p−1}+λ_{0})V +∑_{k=2,...,p−1,p+1,...,p^{2}−1} _{p^{2}}C_{k}w^{p^{2}−k}V^{k}+ ∑_{k=2,...,p−1} _{p}C_{k}w^{p−k}V^{k}. We see that v_{K_{1}}(_{p^{2}}C_{p}w^{p^{2}−p}) = ep + (p^{2} − p)v_{K_{1}}(w) ≥ ep + μ_{1} > μ_{1}, v_{K_{1}}(p^{2}w^{p^{2}−1}) − v_{K_{1}}(pλ_{1}w^{p−1}) = ep + (p^{2} − p)v_{K_{1}}(w)−μ_{0} > 0 and v_{K_{1}}(pλ_{1}w^{p−1})−μ_{0} = ep+μ_{1}+(p−1)v_{K_{1}}(w)−μ_{0} ≥ ep−(μ_{0} −μ_{1}(p+1)/p) > 0. As for the former summation in the expansion above of P (V + w), v_{K_{1}}(_{p^{2}}C_{k}w^{p^{2}−k}) − (p^{2} − k)v(w) > 0. As for the latter summation, the valuation of the coefficient of V^{k} is v_{K_{1}}(_{p}C_{k}w^{p−k}) and v_{K_{1}}(_{p}C_{k}w^{p−k})−(μ_{1}+(μ_{0}−μ_{1})(p−k)/(p−1)) ≥ ep+μ_{1}(p−k)/(p^{2}− p) − (μ_{1} + (μ_{0} − μ_{1})(p − k)/(p − 1)) = ep + (k(pμ_{0} − (p + 1)μ_{1}) + 2pμ_{1} − p^{2}μ_{0})/(p^{2} − p) > ep + (2μ_{1} − pμ_{0})/(p − 1) > 0. Thus P (V + w) has the same Newton polygon as P (W). Then [11, Theorem 4] shows that the affinoid variety splits if and only if j > (pμ_{0} − μ_{1})/(p − 1) and the lemma follows.

In the second case, the Newton polygon of P (W) has no internal vertex. Thus the nonzero roots of P (W) has valuation μ_{0}/(p^{2} − 1). Take one of these roots w and consider the polynomial P (V + w) and its expansion as above. We have v_{K_{1}}(pλ_{1}w^{p−1})−μ_{0} = ep+μ_{1}−pμ_{0}/(p+ 1) > 0 and v_{K_{1}}(p^{2}w^{p^{2}−1}) − μ_{0} = 2ep > 0. Thus the valuation of the coefficient of V in P (V +w) is μ_{0}. Let us show that the valuation of the coefficient of V^{k} is larger than v_{K_{1}}((p^{2}−k)μ_{0}/(p^{2}−1)) = (p^{2}−k)v_{K_{1}}(w) for any k < p^{2}. For k = p, we have μ_{1} ≥ (p^{2} − p)μ_{0}/(p^{2} − 1) and v_{K_{1}}(_{p^{2}}C_{p}w^{p^{2}−p}) > (p^{2} − p)μ_{0}/(p^{2} − 1). As for the former summation in the expansion above, v_{K_{1}}(_{p^{2}}C_{k}w^{p^{2}−k}) − (p^{2} − k)v(w) > 0. As for the latter summaton, v_{K_{1}}(_{p}C_{k}w^{p−k})−(p^{2}−k)v_{K_{1}}(w) = ep−pμ_{0}/(p+1) > 0. Again we see that the Newton polygon of P (V +w) is the same as that of P (W) and the affinoid variety splits if and only if j > p^{2}μ_{0}/(p^{2} − 1). This concludes the lemma.

□

From this lemma, we see that the affinoid variety Y_{M′}^{j} does not split for j ≤ p^{2}(e + m − (r + s))/(p^{2} − 1) and splits if and only if j′ = p(e + m − (r + s)) + j/p satisfies the inequality of the lemma. Thus we have

c(R̃′_{M}) = \{ \begin{matrix}p^{2}(e − r)/(p − 1) + p(r − m) & if m < (ps − r)/(p − 1) \\ p^{2}(e − s)/(p − 1) & if m ≥ (ps − r)/(p − 1). \end{matrix}

Now we consider the affinoid variety

\begin{aligned}X_{M′}^{j}(K̄) = \{ & (x_{1}, x_{2}) ∈ OK̄ × OK̄ | v_{K_{1}}(x_{1}^{p} + π^{e−m}F (π)^{−1}cx_{2}) ≥ j, \\ & v_{K_{1}}(x_{2}^{p^{2}} + π^{e+m−(r+s)}F (π)^{−1}(−c^{−1}x_{1} + dx_{2})) ≥ j \}. \end{aligned}

The map R′_{M} → R̃′_{M} induces the affinoid map f : Y_{M′}^{j} → X_{M′}^{j}. Note that f sends (x, w) to (x, w^{p}) and is surjective. From the proof of [11, Theorem 4], we see that

\begin{aligned}Y_{M′}^{j}(K̄) = ∐_{k=0}^{p^{2}−1}\{ & (x, w) ∈ OK̄ × D(w_{k}, j′ − ∑_{w_{k}≠0} v_{K_{1}}(w_{k})) | \\ & v_{K_{1}}(w^{p^{2}} + π^{e+m−(r+s)}F (π)^{−1}(−c^{−1}x + dw^{p})) ≥ j\} \end{aligned}

for j > c(R̃′_{M}), where j′ = p(e+m−(r +s))+j/p and w_{k}’s is the roots of the polynomial P (W). Let us denote its k-th component by Y_{k}. We claim that f(Y_{k}) ∩ f(Y_{l}) = ∅ for k ≠ l. Suppose that (x, w) ∈ Y_{k} and (x, wζ_{p}^{i}) ∈ Y_{l}. Then v_{K_{1}}(wζ_{p}^{i}−w_{l}) = v_{K_{1}}((w−w_{k})ζ_{p}^{i}+(ζ_{p}^{i}−1)w_{k} +(w_{k} − w_{l})). Now we have v_{K_{1}}((w − w_{k})ζ_{p}^{i}) ≥ j′ − ∑ v_{K_{1}}(w_{k}) > sup v_{K_{1}}(w_{k}) and thus v_{K_{1}}((w − w_{k})ζ_{p}^{i} + (w_{k} − w_{l})) = v_{K_{1}}(w_{k} − w_{l}). If m = (ps − r)/(p − 1), we have v_{K_{1}}(w_{k}) = v_{K_{1}}(w_{k} − w_{l}) for any k ≠ l and therefore v_{K_{1}}(wζ_{p}^{i} − w_{l}) = v_{K_{1}}(w_{k}) < j′ − ∑ v_{K_{1}}(w_{k}), which is a contradiction. Suppose m ≠ (ps − r)/(p − 1). Then, by the notation in the previous lemma, we have v_{K_{1}}(w_{k}−w_{l})−v_{K_{1}}(w_{k}) ≤ (μ_{0}−μ_{1})/(p−1)−μ_{1}/(p^{2}−p), which equals

\{ \begin{matrix}(p(s − m) − (r − m))/(p^{2} − p) & if m < (ps − r)/(p − 1) \\ ((r − m) − p(s − m))/(p − 1) & if m > (ps − r)/(p − 1). \end{matrix}

We see that these values are strictly smaller than v_{K_{1}}(1−ζ_{p}^{i}) = pe/(p− 1) and v_{K_{1}}(wζ_{p}^{i} − w_{l}) = v_{K_{1}}(w_{k} − w_{l}) ≤ sup v_{K_{1}}(w_{k}) < j′ − ∑ v_{K_{1}}(w_{k}). Again this is a contradiction. Therefore we get ♯π_{0}(X_{M′}^{j})K̄ = p^{2}. For j ≤ c(R̃′_{M}), we have ♯π_{0}(X_{M′}^{j})K̄ < p^{2} by the surjectivity of f. Thus c(M′) = c(R̃′_{M}).

Next we prove the assertion on a character. For m = (ps−r)/(p−1), we have s ≡ r mod p − 1 and the I_{K}-module V = G(K̄) contains θ_{K,p−1}^{e−s} = θ_{K,p−1}^{e−r}. Thus we may suppose that m > (ps − r)/(p − 1). By the full faithful theorem of Breuil, it suffices to show that V contains θ_{K,p−1}^{e−s} as an I_{K_{1}}-module. The G_{K_{1}}-set V is identified with the roots of the polynomial Q(X_{2}) = (X_{2}^{p} + λ_{1}X_{2})^{p} + λ_{0}^{p}X_{2} ∈ O_{K_{1}}[X_{2}]. Consider the Newton polygon of Q(X_{2}). For 1 ≤ k ≤ p − 1, the coefficient of X_{2}^{p+(p−1)k} in Q(X_{2}) is _{p}C_{k}λ_{1}^{p−k} and v_{K_{1}}(_{p}C_{k}λ_{1}^{p−k}) − pμ_{1}(p^{2} − (p + (p − 1)k))/(p^{2} − p) = ep > 0. Thus Q(X_{2}) has p − 1 roots of valuation p(μ_{0} − μ_{1})/(p − 1) = p(e − s)/(p − 1) − p(s − m). Put X_{2} = T^{−1}π_{1}^{p(e−s)/(p−1)−p(s−m)}. Then Q(X_{2}) = 0 if and only if T^{p^{2}−1} + a_{0}^{−1}(a_{1}T^{p−1} + π_{1}^{pμ_{0}−(p+1)μ_{1}})^{p} = 0, where a_{k} = λ_{k}/π_{1}^{v_{K_{1}}(λ_{k})}. By Hensel’s lemma, there exists a polynomial R(T) ∈ O_{K_{1}}[T] of degree p − 1, satisfying R(T) ≡ T^{p−1} + t mod π_{1} where t ≠ 0 ∈ k and with the property that R(α) = 0 if and only if α^{−1}π_{1}^{p(e−s)/(p−1)−p(s−m)} is a root of Q(X_{2}) with valuation p(e − s)/(p − 1) − p(s − m). Take such a root w. Then we see that w/π_{1}^{p(e−s)/(p−1)−p(s−m)} ∈ K_{1}^{nr} and g(w) = wg(π_{1}^{p(e−s)/(p−1)})/π_{1}^{p(e−s)/(p−1)} for any g ∈ I_{K_{1}}. By [5, Lemme 3.1.7], we can identify the subspace G(p(e − r))(K̄) of V with the set {0, c^{−1}π_{1}^{p(e−r)/(p−1)}, c^{−1}π_{1}^{p(e−r)/(p−1)}ζ_{p−1}, …, c^{−1}π_{1}^{p(e−r)/(p−1)}ζ_{p−1}^{p−2}}. From the shape of the Newton polygon of Q(X_{2}), we see that w is not contained in this subspace. Therefore the I_{K_{1}}-action on V is tame and thus V = θ_{K_{1},p−1}^{e−s} ⊕ θ_{K_{1},p−1}^{e−r}.

□

In the proof of the theorem, we have shown the following.

Corollary 4.4. If s, m ≥ r, then c(G) = p(e − r)/(p − 1). Otherwise,

c(G) = \{ \begin{matrix}sup(p(e − r)/(p − 1), p(e − s)/(p − 1)) if m ≥ (ps − r)/(p − 1), \\ p(e − r)/(p − 1) + (r − m) if m < (ps − r)/(p − 1). \end{matrix}

References

[1] A. Abbes and A. Mokrane: Sous-groupes canoniques et cycles évanescents padiques pour les variétés abéliennes, Publ. Math. IHES 99 (2004), 117-162

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HASSE-ARF THEOREM FOR 𝔽_{p}-VECTOR SPACE SCHEMES OF RANK TWO23

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