Tame characters and ramification of finite flat group schemes

Shin Hattori

Department of Mathematics, Hokkaido University Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan

∗ Fax: +81-11-706-2636

Email address: shin-h@math.sci.hokudai.ac.jp (Shin Hattori).
URL: http://www.math.sci.hokudai.ac.jp/~shin-h (Shin Hattori).

Preprint submitted to Elsevier Science 9 May 2007

Abstract.

In this paper, for a complete discrete valuation field K of mixed characteristic (0, p) and a finite flat group scheme G of p-power order over O_{K}, we determine the tame characters appearing in the Galois representation G(K̄) in terms of the ramification theory of Abbes and Saito, without any restriction on the absolute ramification index of K or the embedding dimension of G.

Key words. Galois representation; Group scheme; Ramification 1 Introduction

Let K be a complete discrete valuation field of mixed characteristic (0, p) with residue field F which may be imperfect, G_{K} be its absolute Galois group, I_{K} be its inertia subgroup and P_{K} be its wild inertia subgroup. Consider a finite G_{K}-module M of p-power order. Since P_{K} acts unipotently on M, there is a filtration M = M_{0} ⊋ M_{1} ⊋ … M_{m} ⊋ 0 of M by G_{K}-submodules M_{i} such that every graded piece M_{i}/M_{i+1} is tame and killed by p. Then the I_{K}-module M_{i}/M_{i+1} ⊗_{𝔽_{p}} 𝔽̄_{p} is the direct sum of some powers of fundamental characters ([15]). When V comes from a geometrical object, such as a group scheme or the etale cohomology of a proper smooth variety over O_{K}, then these exponents of tame characters often can be described more precisely.

For example, let G be a finite flat group scheme over O_{K} of p-power order. When G is monogenic, that is to say, when the affine algebra of G is generated over O_{K} by one element, it is well-known that the tame characters appearing in the I_{K}-module G(K̄) are determined by the slopes of the Newton polygon of a defining equation of G ([15, Proposition 10]).

On the other hand, for an elliptic modular form f of level N prime to p, we also have a description of the tame characters of the associated mod p Galois representation ρ̄_{f} ([9, Theorem 2.5, Theorem 2.6], [8, Section 4.3]). This is based on Raynaud’s theory of prolongations of finite flat group schemes or the integral p-adic Hodge theory. However, for an analogous study of the associated mod p Galois representation of a Hilbert modular form over a totally real number field, we encounter a local field of higher absolute ramification index. In this case, these two theories no longer work well and we need some other techniques.

In this paper, to propose a new approach toward this problem, we generalize the aforementioned result of [15] to the higher dimensional case using the ramification theory of Abbes and Saito ([2,3]) and determine the tame characters appearing in G(K̄) in terms of the ramification of G without any restriction on the absolute ramification index of K. Namely, we show the following theorem.

Theorem 1.1 Let G be a finite flat group scheme over O_{K}. Write {G^{j}}_{j∈ℚ_{>0}} for the ramification filtration of G in the sense of [2, 3]. Then the graded piece G^{j}(K̄)/G^{j+}(K̄) is killed by p and the I_{K}-module G^{j}(K̄)/G^{j+}(K̄) ⊗_{𝔽_{p}} 𝔽̄_{p} is the direct sum of fundamental characters of level j.

In view of the following corollary, we can regard this theorem as a counterpart for finite flat group schemes of the Hasse-Arf theorem in the classical ramification theory.

Corollary 1.2 Let L be an abelian extension of K. Suppose that there exists a finite flat group scheme G over O_{K} of p-power order such that O_{L} is a Gtorsor over O_{K}. Then the denominator of every jump of the upper numbering ramification filtration {Gal(L/K)^{j}}_{j∈ℚ_{>0}} ([2]) is a power of p.

These results are proved in Section 4 (Theorem 4.7, Theorem 4.8, Corollary 4.10). To prove the main theorem, we firstly show that the tubular neighborhood of G can be chosen to have a rigid analytic group structure. Passing to the closed fiber, we realize the graded piece of the ramification filtration of G as the kernel of an etale isogeny of the additive groups 𝔾̄_{a}^{r} over F̄. Then we determine the tame characters by comparing the I_{K}-action on the graded piece with the 𝔾̄_{m}-action on 𝔾̄_{a}^{r} given by the multiplication. As an appendix, we also give an explicit formula for the conductor of a Raynaud 𝔽-vector space scheme over O_{K} ([14]).

2 Review of the ramification theory of Abbes and Saito

Let K be a complete discrete valuation field with residue field F which may be imperfect. Set π = π_{K} to be an uniformizer of K. The separable closure of K is denoted by K̄ and the absolute Galois group of K by G_{K}. Let 𝔪K̄ and F̄ be the maximal ideal and the residue field of OK̄ respectively. We extend the valuation v_{K} of K to K̄ and normalize this as v_{K}(π) = 1. In [2,3], Abbes and Saito defined the ramification theory of a finite flat O_{K}-algebra of relative complete intersection. Using this, Abbes and Mokrane studied the ramification of finite flat group schemes over O_{K} ([1]). In this section, we gather the necessary definitions and briefly recall their theory.

Let A be a finite flat O_{K}-algebra and 𝔸 be a complete Noetherian semi-local ring (with its topology defined by rad(𝔸)) which is of formally smooth over O_{K} and whose quotient ring 𝔸/rad(𝔸) is of finite type over F. A surjection of O_{K}-algebras 𝔸 → A is called an embedding if 𝔸/rad(𝔸) → A/rad(A) is an isomorphism. A morphism of embeddings (f, f) : (𝔸 → A) → (𝔸′ → A′) is defined to be a pair of O_{K}-homomorphisms f : A → A′ and f : 𝔸 → 𝔸′ which commutes the following diagram.

\begin{matrix}𝔸 & ⟶ & A \\ ^{f}↓ & & ↓^{f} \\ 𝔸′ & ⟶ & A′ \end{matrix}

A morphism (f, f) is said to be finite flat if f is finite flat and the map 𝔸′ ⊗_{𝔸} A → A′ is an isomorphism.

For an embedding (𝔸 → A) and j ∈ ℚ_{>0}, the j-th tubular neighborhood of (𝔸 → A) is the smooth K-affinoid variety X^{j}(𝔸 → A) constructed as follows. Write j = k/l with k, l non-negative integers. Put I = Ker(𝔸 → A) and A_{0}^{k,l} = 𝔸[I^{l}/π^{k}]^{∧}, where ∧ means the π-adic completion. Then A_{0}^{k,l} is a quotient ring of the Tate algebra O_{K}⟨T_{1}, …, T_{r}⟩ for some r. Its generic fiber A_{K}^{j} = A_{0}^{k,l} ⊗_{O_{K}} K is independent of the choice of a representation j = k/l ([3, Lemma 1.4]) and set X^{j}(𝔸 → A) = Sp(A_{K}^{j}). This defines the functor X^{j} from the category of embeddings to the category of smooth K-affinoid varieties, which sends a finite flat map of embeddings to a finite flat map of K-affinoid varieties ([3, Lemma 1.6]).

Note that the functor X^{j} is compatible with the tensor product in the following sense. For embeddings (𝔸 → A) and (𝔹 → B), put ℂ = 𝔸 ˆ⊗_{O_{K}}𝔹 and C = A ⊗_{O_{K}} B. Then (ℂ → C) is also an embedding and we have the canonical isomorphism of K-affinoid varieties

X^{j}(ℂ → C) → X^{j}(𝔸 → A) ×_{K} X^{j}(𝔹 → B).

We put F (A) = Hom_{O_{K}-alg.}(A, OK̄) and F^{j}(A) = \underleftarrow{lim} π_{0}(X^{j}(𝔸 → A)K̄). Here π_{0}(XK̄) denotes the set of geometric connected components of a K-affinoid variety X and the projective limit is taken in the category of embeddings of A. Note that the projective family π_{0}(X^{j}(𝔸 → A)K̄) is constant ([3, Section 1.2]). These define contravariant functors F and F^{j} from the category of finite flat O_{K}-algebras to the category of finite G_{K}-sets. Moreover, there are morphisms of functors F → F^{j} and F^{j′} → F^{j} for j′ ≥ j > 0.

By the finiteness theorem of Grauert and Remmert ([6, Theorem 1.3]), there exists a finite separable extension L of K such that the geometric closed fiber of the unit disc 𝔛^{j}(𝔸 → A)_{O_{L}} for the supremum norm in X^{j}(𝔸 → A)_{L} = X^{j}(𝔸 → A) ×_{K} L is reduced. Then for any finite separable extension L′ of L, the π_{L}-adic formal scheme 𝔛^{j}(𝔸 → A)_{O_{L}} ×_{O_{L}} O_{L′} coincides with the unit disc for the supremum norm in X^{j}(𝔸 → A)_{L′} and thus is normal. The π_{L}- adic formal scheme 𝔛^{j}(𝔸 → A)_{O_{L}} is referred as the stable normalized integral model of X^{j}(𝔸 → A) over L and its geometric closed fiber is denoted by X̄^{j}(𝔸 → A), which is independent of the choice of L. If L/K is Galois, the right action of the Galois group Gal(L/K) on L defined by σ.z = σ^{−1}(z) induces the L-semilinear left action on X^{j}(𝔸 → A)_{L}, which also extends by the functoriality of the unit disc for the supremum norm to the O_{L}-semilinear left action on 𝔛^{j}(𝔸 → A)_{O_{L}}. This is compatible with a base extension of such L and thus defines the canonical F̄-semilinear G_{K}-action on X̄^{j}(𝔸 → A). Then we have the G_{K}-equivariant isomorphism

π_{0}(X̄^{j}(𝔸 → A)) → π_{0}(X^{j}(𝔸 → A)K̄),

where the former is the set of connected components of X̄^{j}(𝔸 → A) ([3, Corollary 1.11]).

Suppose that A is of relative complete intersection over O_{K} and A ⊗_{O_{K}} K is etale over K. Then the natural map F (A) → F^{j}(A) is surjective. The family {F (A) → F^{j}(A)}_{j∈ℚ_{>0}} is separated, exhaustive and its jumps are rational ([2, Proposition 6.4]). The conductor of A is defined to be

c(A) = inf{j ∈ ℚ_{>0} | F (A) → F^{j}(A) is an isomorphism}.

If B is the affine algebra of a finite flat group scheme G over O_{K} which is generically etale, then B is of relative complete intersection (for example, [7, Proposition 2.2.2]) and the theory above can all be applied to B. By the functoriality, F^{j}(B) is endowed with a G_{K}-module structure ([1, Lemme 2.1.1]) and the natural map G(K̄) = F (B) → F^{j}(B) is a G_{K}-homomorphism. Let G^{j} denote the schematic closure ([14]) in G of the kernel of this homomorphism. It is called the j-th ramification filtration of G. We refer c(B) as the conductor of G, which is denoted also by c(G). We put G^{j+}(K̄) = ∪_{j′>j}G^{j′}(K̄) and define G^{j+} to be the schematic closure of G^{j+}(K̄) in G.

As in the classical ramification theory, the ramification filtration {G^{j}}_{j∈ℚ_{>0}} of G is not compatible with closed subgroups but with quotients. Namely, we have the following variant of the classical Herbrand theorem.

Proposition 2.1 ([1, Lemme 2.3.2]) Let G be a finite flat group scheme over O_{K} and H be its closed finite flat subgroup scheme over O_{K}. Then the natural map G^{j}(K̄) → (G/H)^{j}(K̄) is surjective for j ∈ ℚ_{>0}.

In closing this section, we state the following corollary of Proposition 2.1 which is used in Section 4.

Lemma 2.2 Let G be a finite flat group scheme over O_{K} which is generically etale and j > 0 be a jump of the ramification filtration {G^{j}}_{j∈ℚ_{>0}} of G. Then we have c(G/G^{j+}) = j and G^{j}(K̄)/G^{j+}(K̄) = (G/G^{j+})^{j}(K̄).

PROOF. From the definition of j, we see that the subgroup G^{j}(K̄) ⊆ G(K̄) has a non-trivial image in (G/G^{j+})(K̄). By Proposition 2.1, the natural map G^{t}(K̄) → (G/G^{j+})^{t}(K̄) is surjective for any t > 0. We have (G/G^{j+})^{t} = 0 for t > j and (G/G^{j+})^{j} ≠ 0. This concludes the lemma. □

3 Group structure on the tubular neighborhood of a finite flat group scheme

For the rest of the paper, we assume that K is of characteristic 0. Let G = Spf(B) be a connected finite flat group scheme over O_{K}. We define a formal resolution of G to be a closed immersion G → Γ of (profinite) formal group schemes over O_{K}, where Γ = Spf(𝔹) is connected and smooth. Such an immersion can be constructed as follows. By a theorem of Raynaud ([4, Théorème 3.1.1]), we can find an abelian scheme V over O_{K} and a closed immersion of group schemes G → V. Taking the formal completion of V along the zero section, we get a formal resolution of G. We refer the relative dimension of Γ over O_{K} as the dimension of a formal resolution (G → Γ). We define a morphism of formal resolutions to be a pair of group homomorphisms (f, f) which makes the following diagram commutative.

\begin{matrix}G & ⟶ & Γ \\ ^{f}↓ & & ↓^{f} \\ G′ & ⟶ & Γ′ \end{matrix}

Note that a formal resolution of G is also an embedding of B. We say (f, f) is finite flat if this is finite flat in the sense of Section 2. For j ∈ ℚ_{>0}, let us consider the j-th tubular neighborhood X^{j}(𝔹 → B) of the embedding (𝔹 → B), which we also write as X^{j}(G → Γ), or X_{G}^{j} by abuse of notation. Its stable normalized integral model over L and geometric closed fiber are also denoted abusively by 𝔛_{G,O_{L}}^{j} and X̄_{G}^{j}. The following lemma enables us to introduce group structures on these varieties.

Lemma 3.1 Let (𝔸 → A) and (𝔹 → B) be embeddings of finite flat O_{K}- algebras. Put ℂ = 𝔸 ˆ⊗_{O_{K}}𝔹 and C = A⊗_{O_{K}} B. Then the canonical isomorphism

X^{j}(ℂ → C) → X^{j}(𝔸 → A) ×_{K} X^{j}(𝔹 → B)

extends to a canonical isomorphism between their stable normalized integral models

𝔛^{j}(ℂ → C)_{O_{L}} → 𝔛^{j}(𝔸 → A)_{O_{L}} ×_{O_{L}} 𝔛^{j}(𝔹 → B)_{O_{L}}

for any finite extension L over K where these three models are defined.

PROOF. We write the affinoid algebras of X^{j}(𝔸 → A), X^{j}(𝔹 → B) and X^{j}(ℂ → C) as A_{K}^{j}, B_{K}^{j} and C_{K}^{j} respectively. Let Å_{O_{L}}^{j} denote the unit disc in A_{L}^{j} = A_{K}^{j} ˆ⊗_{K}L for the supremum norm. Define B̊_{O_{L}}^{j} and C̊_{O_{L}}^{j} similarly for B and C. From the proof of [6, Theorem 1.3], there exists a continuous surjection α : L⟨T_{1}, …, T_{r′}⟩ → A_{L}^{j} such that ||_{sup} = ||_{α}, where ||_{α} is the residue norm induced by α. We also have a surjection β : L⟨U_{1}, …, U_{s′}⟩ → B_{L}^{j} with the same property for B. Consider the surjection

α ˆ⊗β : L⟨T_{1}, …, T_{r′}⟩ ˆ⊗_{L}L⟨U_{1}, …, U_{s′}⟩ → A_{L}^{j} ˆ⊗_{L}B_{L}^{j} = C_{L}^{j}.

The unit disc in A_{L}^{j} ˆ⊗_{L}B_{L}^{j} for the residue norm induced by α ˆ⊗β is Å_{O_{L}}^{j} ˆ⊗_{π_{L}} B̊_{O_{L}}^{j}, where ˆ⊗_{π_{L}} denotes the π_{L}-adic complete tensor product over O_{L}. Its geometric closed fiber (Å_{O_{L}}^{j} ⊗_{O_{L}} F̄)⊗F̄(B̊_{O_{L}}^{j} ⊗_{O_{L}} F̄) is reduced. By [6, Proposition 1.1], we see that the stable normalized integral model C̊_{O_{L}}^{j} is equal to Å_{O_{L}}^{j} ˆ⊗_{π_{L}} B̊_{O_{L}}^{j}. □

Corollary 3.2 The functor of the j-th tubular neighborhood X^{j} defines a functor from the category of formal resolutions to the category of smooth rigid K-analytic groups. This also induces the two functors below.

(i) The functor defined by (G → Γ) ↦ 𝔛_{G,O_{L}}^{j}, from the full subcategory of the category of formal resolutions which consists of (G → Γ) such that the stable normalized integral model of X_{G}^{j} is defined over L, to the category of smooth π_{L}-adic formal group schemes over O_{L},

(ii) the functor defined by (G → Γ) ↦ X̄_{G}^{j}, from the category of formal resolutions to the category of smooth algebraic groups over F̄.

PROOF. By definition, the canonical isomorphisms in the previous lemma are associative. Hence X_{G}^{j}, 𝔛_{G,O_{L}}^{j} and X̄_{G}^{j} have the group structures induced by that of Γ. The algebraic group X̄_{G}^{j} is reduced, hence smooth by [16, Theorem 11.6]. □

Let G = Spf(B) be a connected finite flat group scheme over O_{K} and (G → Γ = Spf(𝔹)) be a formal resolution of dimension r. Consider the quotient Γ/G as a (profinite) formal group scheme ([10]) and set Γ/G = Spf(𝔸′). We insert here the proof of the following lemma due to the lack of references.

Lemma 3.3 The ring homomorphism of O_{K}-algebras 𝔸′ → 𝔹 is finite flat. Moreover, there is an isomorphism of O_{K}-algebras 𝔸 = O_{K}[[T_{1}, …, T_{r}]] → 𝔸′ which maps the ideal (T_{1}, …, T_{r}) isomorphically to J_{𝔸′} = Ker(𝔸′ → O_{K}).

PROOF. We know from [10, Théorème 1.4] that 𝔸′ is a complete local ring which is a subring of 𝔹, that 𝔹 is isomorphic to ∏_{λ∈Λ} 𝔸′ for some index set Λ and that we have a natural isomorphism 𝔹 ˆ⊗_{𝔸′}𝔹 → B ⊗_{O_{K}} 𝔹. Hence Λ is a finite set and 𝔹 is finite faithfully flat over 𝔸′. By a theorem of Eakin-Nagata ([13, Theorem 3.7]), we see that 𝔸′ is Noetherian. From [13, Theorem 23.7], we see that 𝔸′ ⊗_{O_{K}} F is a regular local ring of dimension r. Since the residue field of 𝔸′ is F, we have an isomorphism F [[T_{1}, …, T_{r}]] → 𝔸′ ⊗_{O_{K}} F which maps the ideal (T_{1}, …, T_{r}) onto the image of J_{𝔸′}. Lifting this map to O_{K}[[T_{1}, …, T_{r}]] → 𝔸′ and using Nakayama’s lemma conclude the proof. □

From this lemma, we can regard the zero section Spf(O_{K}) → Spf(𝔸′) as a formal resolution of the trivial group. We choose once and for all an isomorphism 𝔸 = O_{K}[[T_{1}, …, T_{r}]] → 𝔸′ as in the lemma. The O_{K}-algebra 𝔸 has the formal group law induced by this isomorphism. By definition of 𝔸 and 𝔸′, we have a finite flat map of formal resolutions

G  ⟶ Spf(𝔹)
↓  ↓
Spf(O_{K}) ⟶ Spf(𝔸).

Thus we have a finite flat map of rigid K-analytic groups

f^{j} : X_{G}^{j} = X^{j}(G → Γ) → X^{j}(Spf(O_{K}) → Spf(𝔸)) = D^{r,j},

where D^{r,j} = {(z_{1}, …, z_{r}) ∈ Or ; K̄ | v_{K}(z_{i}) ≥ j for any i}. We call this the affinoid homomorphism associated to a formal resolution (G → Γ). Write B_{K}^{j} and A_{K}^{j} for the K-affinoid algebras of X_{G}^{j} and D^{r,j} respectively. The stable normalized integral model over L of D^{r,j} is denoted by 𝔇_{O_{L}}^{r,j} and its geometric closed fiber by D̄^{r,j}. From Corollary 3.2, we also have a homomorphism

f̊^{j} : 𝔛_{G,O_{L}}^{j} → 𝔇_{O_{L}}^{r,j}

of smooth π_{L}-adic formal group schemes over O_{L} and a homomorphism

f̄^{j} : X̄_{G}^{j} → D̄^{r,j}

of smooth algebraic groups over F̄. We also refer them as the associated homomorphisms to (G → Γ).

Lemma 3.4 The affinoid homomorphism f^{j} : X_{G}^{j} → D^{r,j} is etale for any j > 0. Moreover, for j > c(G), there exists a finite extension K′/K such that X_{G,K′}^{j} is isomorphic to the disjoint sum of finitely many copies of D_{K′}^{r,j}.

PROOF. We have Ω_{B_{K}^{j}/A_{K}^{j}}^{1} = B_{K}^{j} ˆ⊗_{𝔹}Ω̂_{𝔹/𝔸}. It is enough to show that Ω̂_{𝔹/𝔸} is a torsion O_{K}-module. Let J_{𝔸} and J_{𝔹} be the augmentation ideals of 𝔸 and 𝔹 respectively. Set I = Ker(𝔹 → B). Then

Ω̂_{𝔹/𝔸} = Coker(𝔹 ⊗_{𝔸} Ω̂_{𝔸/O_{K}} → Ω̂_{𝔹/O_{K}})

is equal to

𝔹 ⊗_{O_{K}} Coker(Cot(𝔸) → Cot(𝔹)) = 𝔹 ⊗_{O_{K}} J_{𝔹}/(I + J_{𝔹}^{2}) = 𝔹 ⊗_{O_{K}} Cot(B),

where Cot means the cotangent space. This shows the first assertion. For the second assertion, take a finite extension K′ of K where the geometric connected components of X_{G}^{j} are defined. For j > c(G), we have deg(f^{j}) = ♯G(K̄) = ♯π_{0}(X_{G,K′}^{j}) and each of the connected components of X_{G,K′}^{j} is a finite etale cover of D_{K′}^{r,j} whose degree is one. Thus this is isomorphic to D_{K′}^{r,j}. □

Take a finite extension L of K where the stable normalized integral models of X_{G}^{j} and D^{r,j} are defined. The generic fiber G_{L} = G ×_{O_{K}} L can be regarded as a rigid L-analytic subgroup of X_{G,L}^{j} and we have an exact sequence of rigid K-analytic groups

0 → G_{L} → X_{G,L}^{j} → D_{L}^{r,j} → 0.

Let H_{O_{L}}^{j} be the kernel of the associated homomorphism f̊^{j} : 𝔛_{G,O_{L}}^{j} → 𝔇_{O_{L}}^{r,j} as a π_{L}-adic formal group scheme over O_{L}.

Lemma 3.5 The associated homomorphism f̊^{j} : 𝔛_{G,O_{L}}^{j} → 𝔇_{O_{L}}^{r,j} is finite flat. Thus, the π_{L}-adic formal group scheme H_{O_{L}}^{j} can be regarded as a finite flat group scheme over O_{L} and there exists an exact sequence of π_{L}-adic formal group schemes

0 → H_{O_{L}}^{j} → 𝔛_{G,O_{L}}^{j} → 𝔇_{O_{L}}^{r,j} → 0.  (1)

PROOF. From Lemma 3.4, the associated affinoid map f^{j} : X_{G}^{j} → D^{r,j} is finite etale. Let B_{K}^{j} and A_{K}^{j} be their affinoid algebras as above. Let B̊_{O_{L}}^{j} and Å_{O_{L}}^{j} denote their unit disc for the supremum norms. Since D^{r,j} is integral, we see that f^{j} is surjective and the ring homomorphism A_{K}^{j} → B_{K}^{j} is injective. Thus we have an injection Å_{O_{L}}^{j} → B̊_{O_{L}}^{j}, which is finite by [5, Corollary 6.4.1/6]. Hence f̄^{j} : X̄_{G}^{j} → D̄^{r,j} is a surjective homomorphism of algebraic groups over F̄. Since X̄_{G}^{j} and D̄^{r,j} are regular, we see that f̄^{j} is faithfully flat by [13, Theorem 23.1]. Since Å_{O_{L}}^{j} and B̊_{O_{L}}^{j} is π_{L}-torsion free, the map f̊^{j} is flat by the local criterion of flatness. This concludes the lemma. □

From Lemma 3.4, we see that for j > c = c(G), the map f̊^{j} identifies 𝔛_{G,O_{L}}^{j} with the direct sum of finitely many copies of 𝔇_{O_{L}}^{r,j}. More precisely, we have the following.

Lemma 3.6 Let c = c(G) be the conductor of G. Then the associated homomorphism f̊^{j} : 𝔛_{G,O_{L}}^{j} → 𝔇_{O_{L}}^{r,j} is finite etale if and only if j ≥ c.

PROOF. Let ¯0 be the zero section of D̄^{r,j}. Set X_{G}^{j+} = ∪_{j′>j}X_{G}^{j′}. From [3, Lemma 1.12], we have

♯G(K̄) = deg(f̄^{j}) ≥ ♯(f̄^{j})^{−1}(¯0) = ♯π_{0}(X_{G,K̄}^{j+}).

This shows that j ≥ c if and only if f̄^{j} is etale. This is also equivalent to the etaleness of f̊^{j}. □

4 Ramification and the I_{K}-module structure of a finite flat group scheme

Consider the canonical K̄-semilinear left action of G_{K} on X̄_{G}^{j}. When we restrict this to I_{K}, the action is F̄-linear. We call this the geometric monodromy action of I_{K} and write the action of σ ∈ I_{K} as σ_{geom} (note that here we follow the terminology in [2], and not in [3] where the monodromy action is called arithmetic, since in our case there is no “geometric” action other than the monodromy action). Similarly, we have the geometric monodromy action of I_{K} on D̄^{r,j}.

The latter action is described as follows. Let the additive group (resp. multiplicative group) over F̄ be denoted by 𝔾̄_{a} (resp. 𝔾̄_{m}). Consider the left action 𝔾̄_{m} × 𝔾̄_{a}^{r} → 𝔾̄_{a}^{r} given by the multiplication. Write this action of λ ∈ F̄^{×} as [λ]. This action is defined by T_{i} ↦ λT_{i}, where 𝔾̄_{a}^{r} = Spec(F̄[T_{1}, …, T_{r}]). For j ∈ ℚ_{>0}, we define a tame character θ_{j} to be θ_{l′}^{k′}, where k′/l′ is the prime-to-p-denominator part of j mod ℤ ([15]). In other words, we set

θ_{j}(σ) = (σ(π^{1/l′})/π^{1/l′})^{k′} mod 𝔪K̄ ∈ F̄.

Note that, for j = k/l and l = p^{m}l_{0} with (k, l) = 1 and p ∤ l_{0}, we have θ_{j} = θ_{l_{0}}^{kp^{−m}}. We call any of 𝔽_{p}-conjugates of θ_{j} the fundamental character of level j.

Lemma 4.1 There is an isomorphism D̄^{r,j} → 𝔾̄_{a}^{r} of algebraic groups over F̄ such that the geometric monodromy action σ_{geom} on D̄^{r,j} for any σ ∈ I_{K} corresponds by this isomorphism to the multiplication [θ_{j}(σ)] on 𝔾̄_{a}^{r}.

PROOF. Recall that D^{r,j} = X^{j}(𝔸 → O_{K}), where 𝔸 = O_{K}[[T_{1}, …, T_{r}]]. Put j = k/l. Let L be a finite Galois extension of K containing π^{1/l} and e′ = e(L/K) be its ramification index over K. Then e′k/l ∈ ℤ and the affine ring of the stable normalized integral model of D^{r,j} over L is

Å_{O_{L}}^{j} = O_{L}⟨T_{1}/(π_{L})^{e′k/l}, …, T_{r}/(π_{L})^{e′k/l}⟩.

Define an O_{L}-algebra isomorphism O_{L}⟨W_{1}, …, W_{r}⟩ → Å_{O_{L}}^{j} by W_{i} ↦ T_{i}/(π^{1/l})^{k}. Set μ_{𝔸} to be the coproduct of 𝔸 and μ to be the coproduct of the algebra O_{L}⟨W_{1}, …, W_{r}⟩ induced by μ_{𝔸}. We have

μ_{𝔸}(T_{i}) = T_{i} ˆ⊗1 + 1 ˆ⊗T_{i} + (higher degree)

and then μ(W_{i}) is equal to

W_{i} ˆ⊗_{π}1 + 1 ˆ⊗_{π}W_{i} + (π^{1/l})^{k}(higher degree).

This shows the first assertion. In the affine algebra of D̄^{r,j}, we have

σ_{geom}^{∗}(T_{i}/(π^{1/l})^{k}) = T_{i}/σ^{−1}((π^{1/l})^{k}) = θ_{j}(σ)T_{i}/(π^{1/l})^{k}.

This corresponds to the action [θ_{j}(σ)] on 𝔾̄_{a}^{r} = Spec(F̄[W_{1}, …, W_{r}]). □

Next we consider the geometric monodromy action on X̄_{G}^{j} for j ∈ ℚ_{>0}. Let X̄_{G}^{j,0} denote the unit component of the algebraic group X̄_{G}^{j} and H̄^{j} be the geometric closed fiber of H_{O_{L}}^{j}. We begin with the following lemma.

Lemma 4.2 If ψ ∈ End(X̄_{G}^{j,0}) induces the zero map on D̄^{r,j}, then ψ = 0.

PROOF. Put H̄_{0}^{j} = H̄^{j} ∩ X̄_{G}^{j,0}. This is the kernel of the faithfully flat map X̄_{G}^{j,0} → D̄^{r,j} and by assumption we have the following commutative diagram whose rows are exact.

0 ⟶ H̄_{0}^{j} ⟶ X̄_{G}^{j,0} ⟶ D̄^{r,j} ⟶ 0
↓  ^{ψ}↓  ^{0}↓
0 ⟶ H̄_{0}^{j} ⟶ X̄_{G}^{j,0} ⟶ D̄^{r,j} ⟶ 0

Thus ψ factors through H̄_{0}^{j}. Put C̄ = Im(ψ). Then this is a closed subgroup scheme of H̄_{0}^{j} and the map X̄_{G}^{j,0} → C̄ is faithfully flat. Since X̄_{G}^{j,0} is regular and connected, we see that C̄ is also regular and connected by [13, Theorem 23.7]. Hence C̄ = 0 and we have ψ = 0. □

Corollary 4.3 Let G be a connected finite flat group scheme over O_{K}. Take a formal resolution (G → Γ) of dimension r. Then the algebraic group X̄_{G}^{j,0} is isomorphic to 𝔾̄_{a}^{r}.

PROOF. By the previous lemma and Lemma 4.1, we see that X̄_{G}^{j,0} is killed by p. Hence the assertion follows from [12, Lemma 1.7.1]. □

Corollary 4.4 The geometric monodromy action of I_{K} on X̄_{G}^{j,0} is tame.

PROOF. For an element σ of the wild inertia subgroup P_{K}, the geometric monodromy action σ_{geom} on D̄^{r,j} is trivial. Applying the lemma to σ_{geom} −id ∈ End(X̄_{G}^{j,0}) shows the assertion. □

Corollary 4.5 Let J be a finite cyclic quotient of I_{K} through which the tame character θ_{j} factors and τ ∈ J. Let F (t) denote the minimal polynomial of θ_{j}(τ) ∈ 𝔽̄_{p} over 𝔽_{p}. Then the geometric monodromy action of I_{K} on X̄_{G}^{j,0} also factors through J and the equation F (τ_{geom}) = 0 holds in End(X̄_{G}^{j,0}).

PROOF. The first assertion follows from Lemma 4.2 as the previous corollary. The second assertion also follows from this lemma using Lemma 4.1. □

Let c = c(G) be the conductor of G. The lemma below enables us to realize G^{c}(K̄) as a subgroup of X̄_{G}^{c,0}.

Lemma 4.6 The specialization map sp_{c} : X_{G}^{c} → X̄_{G}^{c} induces an I_{K}-equivariant isomorphism G(K̄) → H̄^{c}(F̄) and G^{c}(K̄) → H̄_{0}^{c}(F̄). Here we consider on the left-hand side the natural action as the K̄-valued points of G (resp. G^{c}) and on the right-hand side the restriction of the geometric monodromy action on X̄_{G}^{c}.

PROOF. By definition, the generic fiber of H_{O_{L}}^{c} is equal to G_{L}. From the exact sequence (1) and Lemma 3.6, we know that H_{O_{L}}^{c} is etale over O_{L} and there is the following exact sequence of algebraic groups over F̄.

0 → H̄^{c} → X̄_{G}^{c} → D̄^{r,j} → 0  (2)

Thus we have a natural isomorphism H_{O_{L}}^{c}(K̄) → H̄^{c}(F̄) and the composite

G(K̄) = H_{O_{L}}^{c}(K̄) → H̄^{c}(F̄) → X̄_{G}^{c}(F̄)

coincides with the map sp_{c}. From the isomorphism π_{0}(X̄_{G}^{c}) → π_{0}(X_{G,K̄}^{c}), we see that this map sends G^{c}(K̄) isomorphically onto H̄_{0}^{c}(F̄). For x ∈ X_{G}^{c}(K̄) and σ ∈ I_{K}, let σ(x) denote the natural action of σ on K̄-valued points. Then we have σ_{geom}(x) ∘ σ = σ(x). Taking its specialization shows the I_{K}- equivariance. □

The following theorem can be regarded as a generalization for a finite flat group scheme over O_{K} of the structure theorem of the graded pieces of the classical upper numbering ramification filtration.

Theorem 4.7 Let G be a finite flat group scheme over O_{K} and j ∈ ℚ_{>0}. Then the G_{K}-module G^{j}(K̄)/G^{j+}(K̄) is tame and killed by p.

PROOF. Since G^{j} = (G^{0})^{j}, where G^{0} denotes the unit component of G, we may assume G is connected. From Lemma 2.2, we may also assume j = c = c(G). Take a formal resolution (G → Γ) of dimension r and consider its associated affinoid homomorphism X_{G}^{c} → D^{r,c}. Then the theorem follows from Corollary 4.3, Corollary 4.4, and Lemma 4.6. □

From this theorem, we see that the inertia subgroup I_{K} acts on G^{j}(K̄)/G^{j+}(K̄)⊗_{𝔽_{p}} 𝔽̄_{p} by the direct sum of tame characters. The theorem below determines these characters up to p-power exponent.

Theorem 4.8 Let G be a finite flat group scheme over O_{K} and j ∈ ℚ_{>0}. Then I_{K} acts on G^{j}(K̄)/G^{j+}(K̄) ⊗_{𝔽_{p}} 𝔽̄_{p} by the direct sum of fundamental characters of level j.

PROOF. The same argument as in the proof of Theorem 4.7 reduces the claim to the case where G is connected and j = c = c(G). Take a formal resolution (G → Γ) of dimension r. Let J be as in Corollary 4.5 and τ be a generator of J. Then Corollary 4.5 and Lemma 4.6 show that every eigenvalue of the action of τ_{geom} on the finite dimensional 𝔽_{p}-vector space G^{c}(K̄) is a conjugate of θ_{j}(τ) over 𝔽_{p}. Since the order of J is prime to p, we conclude that I_{K} acts on G^{c}(K̄) ⊗_{𝔽_{p}} 𝔽̄_{p} by the direct sum of 𝔽_{p}-conjugates of θ_{c}. □

Corollary 4.9 Let G be a finite flat group scheme over O_{K}. Then the order of the image of the homomorphism I_{K} → Aut(G(K̄)) is a power of p if and only if every jump j of the ramification filtration {G^{j}}_{j∈ℚ_{>0}} is an element of ℤ[1/p].

PROOF. From Theorem 4.8, we see that the jumps are in ℤ[1/p] if and only if the graded pieces G^{j}(K̄)/G^{j+}(K̄) are unramified. By Theorem 4.7, this is equivalent to the condition that ♯Im(I_{K} → Aut(G(K̄))) is a p-power. □

Corollary 1.2 is the special case of the following result.

Corollary 4.10 Let M be an extension of a complete discrete valuation field over K with p-power relative ramification index e(M/K) and L be an abelian extension of M. Suppose that there exists a finite flat group scheme G over O_{K} of p-power order such that O_{L} is a G-torsor over O_{M}. Then the denominator of every jump of the upper numbering ramification filtration {Gal(L/M)^{j}}_{j∈ℚ_{>0}} is a power of p.

PROOF. We have a natural isomorphism

G ×_{O_{K}} Spec(O_{L}) → Spec(O_{L}) ×_{O_{M}} Spec(O_{L}).

From [1, Lemme 2.1.5] and the assumption on e(M/K), it is sufficient to prove that the jumps of {G^{j}}_{j∈ℚ_{>0}} are contained in ℤ[1/p]. On the other hand, we also have a G_{L}-equivariant isomorphism

Hom_{M-alg.}(L, L̄) = Hom_{L-alg.}(L ⊗_{M} L, L̄) → G(L̄) = G(K̄).

From this, we see that Gal(K̄/L ∩ K̄) acts trivially on G(K̄). Let N be the finite Galois extension of K which corresponds to the kernel of the map G_{K} → Aut(G(K̄)). Since [L : M] is a p-power, [N : M ∩ N] is a p-power. From the assumption on e(M/K), we see that [(M ∩N)^{nr} : K^{nr}] is also a p-power. Hence the image of I_{K} → Aut(G(K̄)) is a p-group and Corollary 4.9 concludes the proof. □

Example 4.11 Let X and Y be proper smooth schemes over O_{K} with geometrically connected fibers. Suppose that X is a G-torsor over Y with some G as in Corollary 4.10. Set L = Frac(Ô_{X,s}) and M = Frac(Ô_{Y,s′}), where s and s′ denote the generic points of the closed fibers of X and Y. Then L and M satisfy the assumption of the corollary.

When G(K̄) is unramified and killed by p, we have the following reinforcement of Corollary 4.9, which is an easy corollary of Proposition 2.1. The author does not know if every jump is an integer whenever G(K̄) is unramified. If G is monogenic, then we see that this holds true from [11, Theorem 4].

Proposition 4.12 Let G be a finite flat group scheme over O_{K} which is killed by p. Suppose that the G_{K}-module G(K̄) is unramified. Then every jump j of the ramification filtration {G^{j}}_{j∈ℚ_{>0}} is an element of pℤ.

PROOF. We may assume K = K^{nr} and G_{K} acts trivially on G(K̄). There is a quotient W of G(K̄)/G^{j+}(K̄) where G^{j}(K̄) has a non-trivial image and of rank one over 𝔽_{p}. Taking the schematic closure of Ker(G(K̄)/G^{j+}(K̄) → W) in G/G^{j+}, we see that W extends to a finite flat group scheme W over O_{K} which is a quotient of G/G^{j+}. By Proposition 2.1, we see that the ramification filtration of W jumps at j. On the other hand, W is a Raynaud 𝔽_{p}-vector space scheme with unramified generic fiber. Thus the proposition follows from [11, Theorem 4]. □

For the rest of this section, we state some corollaries in the case where G is an 𝔽-vector space scheme of rank one or two for a finite extension 𝔽 over 𝔽_{p}. In the case of rank one, Theorem 4.8 directly shows the claim below, while this can be shown also as a corollary of a determination of the conductor of a Raynaud 𝔽-vector space scheme (Theorem 5.5).

Corollary 4.13 Let G be an 𝔽-vector space scheme of rank one over O_{K} and c = c(G). Then the I_{K}-action on the 𝔽-vector space G(K̄) of rank one is given by the fundamental character of level c.

In the case of rank two, we have the following.

Corollary 4.14 Let G be a finite flat 𝔽-vector space scheme of rank two over O_{K} and c = c(G). Then the I_{K}-module G(K̄) ⊗_{𝔽} 𝔽̄_{p} contains the fundamental character of level c. If the G_{K}-module G(K̄) is reducible, this holds true for G(K̄) itself.

PROOF. The first assertion follows easily from Theorem 4.8 and the surjection G^{c}(K̄) ⊗_{𝔽_{p}} 𝔽̄_{p} → G^{c}(K̄) ⊗_{𝔽} 𝔽̄_{p}. Suppose the I_{K}-module G(K̄) is reducible. When G^{c} is of rank one, the assertion is clear from Theorem 4.8. If G^{c} = G, then G^{c} is reducible and the assertion follows also from Theorem 4.8. □

Unlike in the case of tame characters, the values of jumps of {G^{j}}_{j∈ℚ_{>0}} usually carries no information about the extension structure of the I_{K}-module G(K̄), except in some extreme cases such as the corollary below.

Corollary 4.15 Consider an exact sequence of finite flat 𝔽-vector space schemes over O_{K}

0 → G_{1} → G → G_{2} → 0

where G_{1} and G_{2} are connected of rank one. If c(G) = c(G_{2}), then the I_{K}-module G(K̄) splits.

PROOF. Put c = c(G). Take a formal resolution (G → Γ) of dimension r and put Γ_{2} = Γ/G_{1}. Then we get a finite flat map of formal resolutions

\begin{matrix}G & ⟶ & Γ \\ ↓ & & ↓ \\ G_{2} & ⟶ & Γ_{2}. \end{matrix}

Therefore we have a finite flat homomorphism of rigid K-analytic groups X_{G}^{j} → X_{G_{2}}^{j} by Corollary 3.2. As in the proof of Lemma 3.4, we see that this map is finite etale.

Suppose G^{c}(K̄) is of rank one. If G^{c}(K̄) ≠ G_{1}(K̄) as an 𝔽-subspace of G(K̄), the I_{K}-module G(K̄) splits and the proposition follows. Suppose G^{c}(K̄) = G_{1}(K̄). The affinoid variety X_{G}^{c} decomposes to ♯𝔽 components over some finite extension K′ of K. Each component is a Zariski open and closed subset of X_{G,K′}^{c}. As the map f : X_{G,K′}^{c} → X_{G,K′}^{c} is finite etale and X_{G_{2},K′}^{c} is connected, every component X_{G,K′}^{c,i} maps surjectively to X_{G_{2},K′}^{c}. Take some g_{i} ∈ G(K̄) ∩ X_{G,K′}^{c,i}. Using the group structure, we see that

G(K̄) ∩ X_{G,K′}^{c,i} = g_{i} + G^{c}(K̄) = g_{i} + G_{1}(K̄)

and f(G(K̄)∩X_{G,K′}^{c,i}) = f(g_{i}). However, we have f^{−1}(G_{2}(K̄)) = G(K̄) and thus f(G(K̄) ∩ X_{G,K′}^{c,i}) = G_{2}(K̄). This is a contradiction. Therefore we may assume G^{c}(K̄) = G(K̄). In this case, the proposition follows from Theorem 4.7. □

Remark 4.16 Let G be a finite flat group scheme over O_{K} and G^{0} be its unit component. Consider the canonical log structure ℕ → O_{K} defined by 1 ↦ π and a log structure on G over O_{K}. From [3, Lemma 4.2], we see that if the log structure of G is trivial on the generic fiber and G is log flat over O_{K}, then G^{0} is strict over O_{K}. Thus it makes no difference in the description of the tame characters whether we use the non-logarithmic or logarithmic ramification theory of [2, 3].

5 Example: 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 5.4 below.

Let K be a complete discrete valuation field of mixed characteristic (0, p). We write e for its absolute ramification index. For a ∈ K̄ and j ∈ ℚ_{>0}, 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 ([14]) over O_{K} defined by the r equations

T_{1}^{p} = π^{s_{1}}T_{2}, T_{2}^{p} = π^{s_{2}}T_{3}, …, T_{r}^{p} = π^{s_{r}}T_{1}.

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 5.1 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

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 shows that 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 5.2 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 5.3 For g_{1}(Y_{1}, …, Y_{d}), g_{2}(Y_{1}, …, Y_{d}) ∈ K[Y_{1}, …, Y_{d}] and j_{1} ≥ j_{2}, we have an equality of affinoid varieties

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

= {(x, y) ∈ OK̄ × Od ; K̄ | v_{K}(x − g_{1}(y)) ≥ j_{1}, v_{K}(g_{1}(y) − g_{2}(y)) ≥ j_{2}}.

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

Lemma 5.4 Let a ∈ O_{K} and s = v_{K}(a). If j ≤ pe/(p − 1) + s, then the affinoid variety X^{j}(K̄) = {(x, y) ∈ O2 ; K̄ | v_{K}(x^{p} − ay^{p^{n}}) ≥ j} is equal to

{(x, y) ∈ O2 ; K̄ | v_{K}(x − a^{1/p}y^{p^{n−1}}) ≥ j/p}.

PROOF. Lemma 5.1 shows that the fiber of the second projection X^{j}(K̄) → OK̄ at y is equal to

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.

Thus we have

X^{j}(K̄) = {(x, y) ∈ O2 ; K̄ | 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 5.5 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 the group scheme is etale and the conductor is 0. Thus we may assume j_{r} > 0 and G(s_{1}, …, s_{r}) is connected. Let A be the affine algebra of this group scheme and put B = A[W]/(W^{p^{r−1}} − T_{1}). Consider the finite flat map of embeddings

O_{K}[[T_{1}, …, T_{r}]] ⟶ A
↓  ↓
O_{K}[[W, T_{2}, …, T_{r}]] ⟶ B

where the left vertical arrow is defined by T_{1} ↦ W^{p^{r−1}}. This induces a finite flat map 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}

be the j-th tubular neighborhoods of these two embeddings. We calculate a jump of {F^{j}(B)}_{j∈ℚ_{>0}} at first.

Lemma 5.6 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 5.6 and Theorem 5.5, 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 5.4 and Lemma 5.3 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 5.2, 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 5.6. □

Lemma 5.7 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′−1} 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,K̄}^{j}) → π_{0}(X_{A,K̄}^{j}) 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 5.4 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). □

Acknowledgements

The author would like to thank Takeshi Saito for his encouragements and valuable discussions. He also wants to thank Ahmed Abbes and Takeshi Tsuji for their helpful comments, and the anonymous referee for a number of suggestions improving the paper.

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