Congruences of Néron models for tori and the Artin conductor

Ching-Li Chai^{1} and Jiu-Kang Yu^{2}

National Center for Theoretical Science National Tsing-Hua University, Hsinchu, Taiwan

with an appendix

by Ehud de Shalit

Institute of Mathematics, Hebrew University Giv’at-Ram, Jerusalem 91904 ISRAEL

Version 2.1, September 30, 1999

§1. Introduction

Let K be a complete local field with a perfect residue field. Then every torus T over K has a Néron model \underline{T}^{NR} over the ring of integers 𝒪 of K. The standard reference for Néron models is the book [BLR]. The torus is canonically determined by the lattice Λ = X_{∗}(T) and the action of Γ = Gal(K^{sep}/K) on Λ. Therefore, \underline{T}^{NR} depends only on (K, Λ).

For large e, the action of Γ on Λ factors through the quotient Γ(Tr_{e} K) of Γ. Here, Γ(Tr_{e} K) is the Galois group classifying extensions of K that are “at most e-ramified”, and it depends only on Tr_{e} K via the theory of Deligne [D], where Tr_{e} K = (𝒪/𝔭^{e}, 𝔭/𝔭^{e+1}, ϵ) is Deligne’s e^{th} truncation (𝔭 being the prime ideal of 𝒪, and ϵ being the canonical map from 𝔭/𝔭^{e+1} to 𝒪/𝔭^{e}).

What does (Tr_{e} K, Λ) tell us about \underline{T}^{NR}? It is natural to expect that (Tr_{e} K, Λ) determines \underline{T}^{NR} ⊗(𝒪/𝔭^{N}) when e ≫ N. The main result (9.2) of this paper confirms this expectation. Another version (8.5) of our main result is formulated without Deligne’s theory, and is valid for local fields that are not necessarily complete.

This result leads to a solution of a question of Gross, which was communicated to us by G. Prasad and motivated this work. The Néron model does not commute with base field extension before the torus splits. Choose a finite Galois extension L/K which splits T. One can measure the change of Néron models when passing from K to L by comparing (Lie \underline{T}^{NR}) ⊗ 𝒪_{L} with Lie(\underline{(T ⊗ L)}^{NR}). The length of the 𝒪_{L}-module

\frac{Lie(\underline{(T ⊗ L)}^{NR})}{(Lie \underline{T}^{NR}) ⊗ 𝒪_{L}}

divided by the ramification index of L/K is an invariant c(T) of the torus T. The question is to understand this invariant. An argument in Gross and Gan [GG] shows that if it is known that c(T) depends only on the isogeny class of T (i.e. only on Λ ⊗ ℚ) for all T over K, then c(T) is equal to one-half of the Artin conductor of the Galois representation Λ ⊗ ℚ.

This invariance under isogeny is not difficult to check when the residue field of 𝒪 is of characteristic 0. The case when K is a finite extension of ℚ_{p} can be proved using Tate’s formula for the Euler-Poincaré characteristic for Galois cohomologies of local fields, and was known to Chai, Gross, and Kushnirsky independently. This argument can be extended to handle any local field of characteristic 0 with perfect residue field, see §11.

1partially supported by grant DMS 9800609 from the National Science Foundation

2partially supported by grant DMS 9801633 from the National Science Foundation

When K is of equal-characteristic p, the above method doesn’t work and the problem is a major obstruction for extending Gross’s theory [G] of the motive of a reductive group to the function field setting in the work of Kushnirsky [K]. Using the main result of this paper, together with Deligne-Kazhdan-Krasner’s idea that local fields in characteristic p are limits of local fields of characteristic 0 as the absolute ramification index tending to infinity, we solve the problem in this case.

E. de Shalit has used this technique to obtain interesting arithmetic results [dS]. During the writing of this work, we learned that he has found a very nice solution to Gross’s question, also based on Deligne-Kazhdan-Krasner’s idea. His method deals with the Lie algebras directly and is different from ours. He has kindly written up his solution as an appendix to this paper. Together with the definition of the invariant c(T) and Theorem 11.3, the appendix provides a complete proof that c(T) is equal to one-half of the Artin conductor of Λ ⊗ ℚ.

We now give some indications about the proof of the main result. For simplicity, assume that 𝒪 is strictly henselian for the rest of this introduction. Recall that the Néron model is constructed by starting with a suitable model \underline{T}^{0}, and then performing a series of dilatations to \underline{T}^{0}. When one tries to understand the reduction of the Néron model from this construction, several difficulties arise: (i) it is not clear whether \underline{T}^{0} ⊗ (𝒪/𝔭^{N}) depends only on the truncated data (Tr_{e} K, Λ); (ii) in general it is difficult to relate the reduction of a model before and after doing a dilatation, unless the model satisfies some strong regularity condition, e.g. smoothness; (iii) the center of (the first) dilatation is the image of \underline{T}^{0}(𝒪) → \underline{T}^{0}(𝒪/𝔭), which is not a priori determined by the truncated data only.

(iii) is resolved by using a result of Elkik, which tells us that the image of \underline{T} (𝒪) → \underline{T} (𝒪/𝔭) is the same as the image of \underline{T} (𝒪/𝔭^{N}) → \underline{T} (𝒪/𝔭) under suitable conditions. This type of result also allows us to handle (i) as follows: we find a (non-flat) model \underline{T}′ ⊃ \underline{T}^{0} whose reduction depends only on the truncated data, then we show that \underline{T}^{0} ⊗ (𝒪/𝔭^{N}) can be recovered inside \underline{T}′ ⊗ (𝒪/𝔭^{N}) as the schematic closure of a suitable collection of points coming from reduction modulo π^{N}, which can be controlled by Elkik’s result again.

We overcome (ii) by embedding \underline{T}^{0} into a smooth model \underline{R} of a larger torus, and perform the dilatations to \underline{R} instead. At the end, we need to recover \underline{T}^{NR} as the schematic closure of T in the dilatation of \underline{R}. This is similar to the situation of recovering \underline{T}^{0} inside \underline{T}′, and is handled by an analogous method. There is another way to deal with (ii) by using Elkik’s result, but our present exposition may be psychologically neater.

We thank Professor G. Prasad for bringing the question of Gross to our attention, explaining to us its applications in the context of [G] and [P], and his encouragement. We also thank Prof. E. de Shalit for allowing his work to appear as an appendix. Discussions with P. Deligne. W.T. Gan, and B. Gross are gratefully acknowledged. We are also grateful for the support and hospitality of the National Center for Theoretical Science, Hsinchu, Taiwan, where most of this work was done during our visit in the summer of 1999.

§2. Notations

(2.1) Let 𝒪 = 𝒪_{K} be a discrete valuation ring with residue field κ and let K be its field of fractions. Let π = π_{K} be a prime element of 𝒪. The strict henselization and the completion of 𝒪 are denoted by 𝒪^{sh} and 𝒪̂ respectively. Their fields of fractions are denoted by K^{sh} and K̂ respectively. The residue field of 𝒪^{sh} is denoted by κ^{sep}, as it is a separable closure of κ.

(2.2) Let T be a torus over K, and L/K be a Galois extension such that T is split over L. Put Γ = Gal(L/K) and Λ = X_{∗}(T), the cocharacter group of T. Then T is determined by the Γ-module Λ up to a canonical isomorphism.

(2.3) We will occasionally work with another discrete valuation ring 𝒪_{0}. We denote analogous constructs by the same notations with a subscript 0. We introduce a series of congruence notations:

– (𝒪, 𝒪_{L}) ≡_{α} (𝒪_{0}, 𝒪_{L_{0}}) (level N): this means that α is an isomorphism from 𝒪_{L}/π^{N}𝒪_{L} to 𝒪_{L_{0}}/π_{0}^{N}𝒪_{L_{0}} and induces an isomorphism 𝒪/π^{N}𝒪 → 𝒪_{0}/π_{0}^{N}𝒪_{0}.

– (𝒪, 𝒪_{L}, Γ) ≡_{α,β} (𝒪_{0}, 𝒪_{L_{0}}, Γ_{0}) (level N): this means (𝒪, 𝒪_{L}) ≡_{α} (𝒪_{0}, 𝒪_{L_{0}}) (level N), β is an isomorphism Γ → Γ_{0}, and α is Γ-equivariant relative to β: α(γ.x) = β(γ).α(x).

– (𝒪, 𝒪_{L}, Γ, Λ) ≡_{α,β,ϕ} (𝒪_{0}, 𝒪_{L_{0}}, Γ_{0}, Λ_{0}) (level N): this means that (𝒪, 𝒪_{L}, Γ) ≡_{α,β} (𝒪_{0}, 𝒪_{L_{0}}, Γ_{0}) (level N), and ϕ is isomorphism Λ → Λ_{0} which is Γ-equivariant relative to β.

– If it is not necessary to name the isomorphisms (α, β, etc.), we omit them from the notation.

(2.4) In this paper, “X is determined by (𝒪/π^{N}𝒪, 𝒪_{L}/π^{N}𝒪_{L}, Γ, Λ)” means: if

(𝒪, 𝒪_{L}, Γ, Λ) ≡_{α,β,ϕ} (𝒪_{0}, 𝒪_{L_{0}}, Γ_{0}, Λ_{0}) (level N),

then there is a canonical isomorphism X → X_{0} determined by (α, β, ϕ) (when X is a scheme), or there is a canonical isomorphism determined by (α, β, ϕ) sending X to X_{0} (when X is a function, or a point, etc).

(2.5) All rings in this paper are 𝒪-algebras or 𝒪_{0}-algebras. If R is an 𝒪-algebra, we denote the ring R/π^{N}R by R^{N}. We do the same for 𝒪_{0}-algebras.

§3. The construction of the Néron models of a torus

(3.1) Néron models We have three notions of Néron models of T.

– The lft Néron model \underline{T}^{lft NR} as defined in [BLR, Ch. 10], satisfying \underline{T}^{lft NR}(𝒪^{sh}) = T (K^{sh}).

– The smooth model \underline{T}^{ft NR} with connected generic fiber such that \underline{T}^{ft NR}(𝒪^{sh}) is the maximal bounded subgroup of T (K^{sh}). This model is of finite type over 𝒪.

– The connected smooth model \underline{T}^{conn NR} such that \underline{T}^{conn NR}(𝒪^{sh}) is the Iwahori subgroup of T (K^{sh}). This model is the neutral component of either of the previous two models. It has connected generic fiber and special fiber, and is of finite type over 𝒪.

It is most convenient for us to work with the second notion. So let \underline{T}^{NR} = \underline{T}^{ft NR}.

(3.2) Construction We now recall the construction of the Néron model of T as explained in [BLR]. Let \underline{T}^{0} be any group scheme over 𝒪 of finite type such that \underline{T}^{0}(𝒪^{sh}) = \underline{T}^{NR}(𝒪^{sh}) (recall that \underline{T}^{NR}(𝒪^{sh}) is an intrinsically defined subset of T (K^{sh})). Then \underline{T}^{NR} can be obtained by applying the smoothening process to \underline{T}^{0}. The process produces a series of models \underline{T}^{i}, as follows:

Let Z^{i} be the Zariski closure of \{x mod π ∈ \underline{T}^{i}(κ^{sep}) : x ∈ \underline{T}^{i}(𝒪^{sh})\}, as a closed subscheme of \underline{T}^{i} ⊗ κ with the reduced induced structure. Then \underline{T}^{i+1} is the dilatation of Z^{i} on \underline{T}^{i}.

Finally, \underline{T}^{NR} = \underline{T}^{i} for i ≫ 0. More precisely, if δ = max\{δ(x) : x ∈ T^{0}(𝒪^{sh})\}, then \underline{T}^{NR} = \underline{T}^{i} for all i ≥ δ. Here δ(x) is Néron’s measure for the defect of smoothness. See §5 for more discussions.

(3.3) One way to show the existence of \underline{T}^{0} is to take R to be a torus containing T such that \underline{R}^{NR} is easy to describe. For example, we can take R to be an induced torus (we will do so in a canonical way later). Let \underline{T}^{0} be the schematic closure of T in \underline{R}^{NR}. Then \underline{T}^{0}(𝒪^{sh}) = \underline{T}^{NR}(𝒪^{sh}).

There are other ways to achieve this. For example, each \underline{T}^{i} can serve as “\underline{T}^{0}”, because \underline{T}^{i}(𝒪^{sh}) = \underline{T}^{i−1}(𝒪^{sh}) by construction. Notice that \underline{T}^{i} is flat over 𝒪 by construction for all i ≥ 1. But we don’t have to start with a flat \underline{T}^{0}.

(3.4) For our purpose, it is convenient to reformulate the above slightly differently. We maintain our assumptions about R and \underline{R}^{NR}. But we only assume that \underline{T}^{0} is a closed subgroup of \underline{R}^{NR} with generic fiber T such that \underline{T}^{0}(𝒪^{sh}) = \underline{T}^{NR}(𝒪^{sh}).

We first produce a sequence of smooth models \underline{R}^{i} of R as follows:

Let \underline{R}^{0} = \underline{R}^{NR}. For i ≥ 0, let W^{i} be the Zariski closure of \{x mod π ∈ \underline{R}^{i}(κ^{sep}) : x ∈ \underline{T}^{0}(𝒪^{sh}) ⊂ \underline{R}^{i}(𝒪^{sh})\}, as a subscheme of \underline{R}^{i}⊗κ with the reduced induced structure. Then \underline{R}^{i+1} is the dilatation of W^{i} on \underline{R}^{i}. Notice that by the defining properties of the dilatation, the inclusion \underline{T}^{0}(𝒪^{sh}) ⊂ \underline{R}^{i}(𝒪^{sh}) induces an inclusion \underline{T}^{0}(𝒪^{sh}) ⊂ \underline{R}^{i+1}(𝒪^{sh}), which justifies this inductive definition.

(3.5) Lemma The schematic closure of T in \underline{R}^{i} is \underline{T}^{i} for all i ≥ 1. In particular, it is \underline{T}^{NR} for i ≫ 0.

Proof. We proceed by induction. By induction hypothesis, \underline{T}^{i−1} is a closed subgroup of \underline{R}^{i−1}, and W^{i−1} is the image of Z^{i−1} in \underline{T}^{i−1} → \underline{R}^{i−1}. It follows from [BLR, 3.2/Prop. 2(cd)] that \underline{T}^{i} is a closed subgroup of \underline{R}^{i}. It follows that \underline{T}^{i} is the schematic closure of its generic fiber T in \underline{R}^{i}, because \underline{T}^{i} is flat.

(3.6) As mentioned earlier, we may and will always choose the torus R canonically: we just take R = Res_{L/K}(T ⊗ L) (Weil’s restriction). Notice that T ×_{Spec K} Spec L is split and canonically isomorphic to Λ ⊗_{ℤ} 𝔾_{m}. Therefore, \underline{R}^{NR} is simply Λ ⊗_{ℤ} (Res_{𝒪_{L}/𝒪} (𝔾_{m}_{/𝒪_{L}} )).

Let \underline{T}^{0} be the schematic closure of T in \underline{R}^{NR} and put \underline{T} _{L}^{0} = \underline{T}^{0} ⊗ 𝒪_{L}, the schematic closure of T ⊗ L in \underline{R}^{NR} ⊗ 𝒪_{L}. We will also need to consider the following group schemes over 𝒪_{L}: let \underline{R}′ = \underline{R}^{NR} ⊗𝒪_{L}, \underline{R}^{†} = X_{∗}(R ⊗_{K} L)⊗_{ℤ}(𝔾_{m}_{/𝒪_{L}} ), \underline{T}^{†} = X_{∗}(T ⊗_{K} L)⊗_{ℤ}(𝔾_{m}_{/𝒪_{L}} ). There are canonical morphisms \underline{T}^{†} → \underline{R}^{†}, φ : \underline{R}′ → \underline{R}^{†}. Let \underline{T}′ = \underline{T}^{†} ×_{\underline{R}}_{†} \underline{R}′. So we have a Cartesian diagram

\begin{matrix}\underline{T}′ & ⟶ & \underline{T}^{†} \\ ↓ & & ↓ \\ \underline{R}′ & ⟶^{φ} & \underline{R}^{†} \end{matrix}

The vertical arrows are closed immersions. Since \underline{T}′ has generic fiber T ⊗ L, \underline{T} _{L}^{0} is equal to the schematic closure of T ⊗ L in \underline{T}′.

Observe that the whole diagram modulo π^{N} is determined by (𝒪^{N}, 𝒪_{L}^{N}, Γ, Λ). This remark will be useful later in (5.5) and (8.2.2).

(3.7) Remark Since we choose \underline{T}^{0} canonically, it follows that the model \underline{T}^{i} is canonically associated to T_{/K} for each i. Our methods also show that the reduction of \underline{T}^{i} depends only on the truncated data of sufficiently high level.

§4. Dilatation of smooth schemes and reduction

(4.1) As mentioned in the introduction, a difficulty in studying the reduction of \underline{T}^{i} is the following: if A is the affine ring of \underline{T}^{i} and Z^{i} is defined by the ideal (π, f_{1}, …, f_{n}), then the affine ring of \underline{T}^{i+1} is

A[X_{1}, …, X_{n}]/(πX_{1} − f_{1}, …, πX_{n} − f_{n}) mod π^{∞}-torsion.

We don’t know how to control the above torsion, not even locally.

However, we have the following:

(4.2) Proposition Let X be a smooth scheme over 𝒪, and W be a closed smooth subscheme over X ⊗ κ. Let X′ be the dilatation of W on X. Then for any N ≥ 1, X′ ⊗ 𝒪^{N} depends only on X ⊗ 𝒪^{N+1} in a canonical way.

(4.2.1) Canonicity We first explain the canonicity. Recall that X′ is an open subscheme of the blow-up Bl(X, W) of X with center W, which is defined to be

Bl(X, W) = Proj ⨁_{t≥0} ℐ^{t},

where ℐ^{t} is the t^{th} power of the ideal sheaf ℐ of W in 𝒪_{X}. On the other hand, Bl(X, W) is a closed subscheme of Bl′(X, ℐ), where

Bl′(X, ℐ) = Proj ⨁_{t≥0} Sym_{𝒪_{X}}^{t} ℐ.

The closed immersion Bl(X, W) → Bl′(X, ℐ) is induced by the natural morphism from the symmetric power Sym^{t} ℐ to ℐ^{t}.

While the dilatation is defined only in a very special context and the blow-up doesn’t behave well under base change, Bl′(X, ℐ) can be defined whenever ℐ is a coherent sheaf of 𝒪_{X}-module, and its formation clearly commutes with arbitrary base change. It follows that Bl′(X, ℐ) ⊗ 𝒪^{N} = Bl′(X ⊗ 𝒪^{N}, ℐ ⊗ 𝒪^{N}) depend only on (X ⊗ 𝒪^{N+1}, W), because π𝒪_{X} ⊂ ℐ ⊂ 𝒪_{X}.

Now assume that X_{1} and X_{2} are 𝒪-schemes, and ϕ is an isomorphism X_{1} ⊗𝒪^{N+1} → X_{2} ⊗𝒪^{N+1}. Assume also that W_{1} ⊂ X_{1} ⊗κ, W_{2} ⊂ X_{2} ⊗κ are closed subschemes smooth over κ and ϕ induces an isomorphism from W_{1} to W_{2}. Form the dilatation X′_{i} and Y_{i} = Bl′(X_{i}, ℐ_{i}), i = 1, 2. Our canonicity statement is: the natural isomorphism Bl′(ϕ) : Y_{1} ⊗ 𝒪^{N} → Y_{2} ⊗ 𝒪^{N} induces an isomorphism from the subscheme X′_{1} ⊗ 𝒪^{N} of Y_{1} ⊗ 𝒪^{N} to X′_{2} ⊗ 𝒪^{N}.

(4.2.2) Proof of Prop. (4.2) The following local calculation is taken from [BLR, 3.2/Prop. 3]. Let i = 1, 2. Let x′_{i} be a point on X′_{i} ⊗ κ which projects to x_{i} ∈ X_{i} ⊗ κ. The local calculation there shows that we can choose a system of local coordinates f_{1}^{(i)}, …, f_{r}^{(i)}, g_{r+1}^{(i)}, …, g_{n}^{(i)} at x_{i} on X_{i}_{/𝒪} such that W^{i} is defined by (π, g_{r+1}^{(i)}, …, g_{n}^{(i)}) near an affine neighborhood U_{i} of x_{i}, X′_{i} above U_{i} is simply

Spec(B′_{i}/π^{∞}-torsion), where B′_{i} = 𝒪_{X_{i}}(U_{i})[Y_{r+1}^{(i)}, …, Y_{n}^{(i)}]/(πY_{r+1}^{(i)} − g_{r+1}^{(i)}, …, πY_{n}^{(i)} − g_{n}^{(i)}),

and f_{1}^{(i)}, …, f_{r}^{(i)}, Y_{r+1}^{(i)}, …, Y_{n}^{(i)} form a system of local coordinates at x′_{i} on X′_{i}_{/𝒪}.

It is seen directly that (B′_{i})_{x′_{i}} is regular local ring, hence free of π^{∞}-torsion. It follows that a suitable localization (B′_{i})_{h} is already free of π^{∞}-torsion. By shrinking U, we may assume that B′_{i} is free of π^{∞} torsion.

If ϕ(x_{1}) = x_{2}, we can arrange to have: ϕ^{∗}(f_{j}^{(2)} mod π^{N}) ≡ f_{j}^{(1)} mod π^{N}, ϕ^{∗}(g_{k}^{(2)} mod π^{N}) ≡ g_{k}^{(1)} mod π^{N}, and ϕ induces an isomorphism ϕ̃^{∗} : 𝒪_{X_{2}}(U_{2}) ⊗ 𝒪^{N} → 𝒪_{X_{1}}(U_{1}) ⊗ 𝒪^{N}. Then it is clear that we have an isomorphism (ϕ′)^{∗} : B′_{2} ⊗ 𝒪^{N} → B′_{1} ⊗ 𝒪^{N} which extends (ϕ̃)^{∗} and sends Y_{k}^{(2)} to Y_{k}^{(1)}.

It remains to show that ϕ′ is indeed induced by Bl′(ϕ). Above U_{i} ⊗ 𝒪^{N}, the affine ring of Bl′(X_{i}, ℐ) ⊗ 𝒪^{N} is just

B′′_{i} = (⨁_{t≥0} Sym_{B_{i}^{N}}^{t} I_{i}^{N})_{π_{1}}^{deg 0}

where B_{i}^{N} = 𝒪_{X_{i}}(U_{i}) ⊗ 𝒪^{N}, I_{i}^{N} = (π, g_{r+1}^{(i)}, …, g_{n}^{(i)}) ⊗ 𝒪^{N}, the subscript π_{1} indicates localizing regarding π as a homogeneous element of degree 1, and the superscript deg 0 indicates taking the degree 0 part of the localized ring.

The ring B′′_{i} maps to (B′_{i})^{N} by sending π_{1}^{−1}g_{k}^{(i)} to Y_{k}. Now it is clear that (ϕ′)^{∗} is indeed induced by Bl′(ϕ)^{∗} : B′′_{2} → B′′_{1}. The lemma is proved.

(4.3) Corollary For i ≥ 0, N ≥ 1, \underline{R}^{i+1} ⊗ 𝒪^{N} depends only on \underline{R}^{i} ⊗ 𝒪^{N+1} in a canonical way.

Proof. Because W^{i} is a reduced algebraic group over κ, it is smooth over κ. Therefore, we can apply the proposition.

§5. Bounding the defect of smoothness

(5.1) Change of base field Let X be any scheme of finite type over 𝒪 such that X ⊗K is smooth over K. Consider x ∈ X(𝒪^{sh}) as a morphism Spec 𝒪^{sh} → X. Recall that Néron’s measure of the defect of smoothness δ(x) is defined by

δ(x) = δ(x; X) = length_{𝒪^{sh}} torsion(x^{∗}Ω_{X/𝒪}^{1} ),

the length of the torsion part of x^{∗}Ω_{X/𝒪}^{1} . If we consider x as a point of X ⊗𝒪_{L}, then δ(x; X ⊗𝒪_{L}) = e(L/K) · δ(x; X), where e(L/K) is ramification index of L/K. This follows from the elementary fact that the formation of Ω^{1} commutes with base change.

(5.2) Closed immersion Let i : X ⊂ X′ be a closed immersion of 𝒪-schemes such that i induces an isomorphisms X ⊗K → X′ ⊗K. Then we have a surjection i^{∗}Ω_{X′/𝒪}^{1} → Ω_{X/𝒪}^{1} . Therefore, for any x ∈ X(𝒪^{sh}), we have a surjection (i ∘ x)^{∗}Ω_{X′/𝒪}^{1} → x^{∗}Ω_{X/𝒪}^{1} . It follows that δ(x; X) ≤ δ(i ∘ x; X′).

(5.3) Group scheme Suppose that X is a group scheme over 𝒪 and e ∈ X(𝒪^{sh}) is the identity element. For each x ∈ X(𝒪^{sh}), let r_{x} : X⊗𝒪^{sh} → X⊗𝒪^{sh} be the isomorphism of right multiplication by x. Then r_{x} ∘ e = x and hence e^{∗}Ω_{X/𝒪^{sh}}^{1} = x^{∗}Ω_{X/𝒪^{sh}}^{1} . Therefore, δ(e) = δ(x). We conclude that in (3.2), it suffices to take i ≥ δ(e).

(5.4) We would like to control δ(e; \underline{T}^{0}). By (5.1), (5.2), and (3.6), we have

δ(e, \underline{T}^{0}) ≤ \frac{δ(e, \underline{T}′)}{e(L/K)}.

So it is enough to control δ(e; \underline{T}′). According to [BLR, 3.3], δ(e; \underline{T}′) can be calculated by embedding \underline{T}′ into a smooth scheme over 𝒪_{L} as a closed subscheme. We use the embedding \underline{T}′ → \underline{R}′.

The schemes \underline{T}^{†} and \underline{R}^{†} can be written very explicitly. In fact \underline{T}^{†} is cut out by nd − d explicit equations f_{1}, …, f_{nd−d} on \underline{R}^{†}, where d = dim T, n = [L : K]. It follows that \underline{T}′ is cut out by the equations φ^{∗}f_{1}, …, φ^{∗}f_{nd−d} on \underline{R}′.

Now let z_{1}, …, z_{nd} be a system of local coordinates near e, and put

M = (\frac{∂(φ^{∗}f_{i})}{∂z_{j}})_{{1≤i≤nd−d \atop 1≤j≤nd}} .

Then δ(e; \underline{T}′) is the minimum of the valuation of e^{∗}Δ, for all (nd − d)-minors Δ of M.

(5.5) Lemma Suppose that (𝒪, 𝒪_{L}, Γ, Λ) ≡ (𝒪_{0}, 𝒪_{L_{0}}, Γ_{0}, Λ_{0}) (level N) with Ne(L/K) > δ(e; \underline{T}′). Form \underline{T} ′_{0} in the same way that we form \underline{T}′. Then δ(e; \underline{T} ′_{0}) = δ(e; \underline{T}′)

Proof. For any N ≥ 1, all the following objects are clearly determined by (𝒪^{N}, 𝒪_{L}^{N}, Γ, Λ) only (see the end of (3.6)): \underline{R}^{†} ⊗ 𝒪_{L}^{N}, \underline{T}^{†} ⊗ 𝒪_{L}^{N}, \underline{R}′ ⊗ 𝒪_{L}^{N}, \underline{T}′ ⊗ 𝒪_{L}^{N}, the matrix e^{∗}(M mod π^{N}), min(δ(e; \underline{T}′), Ne(L/K)). Now the lemma is obvious.

§6. Singularities of commutative group schemes

In this section, we assemble some known facts about relative complete intersection (r.c.i.) morphisms to be used later. The basic reference is [EGA IV, 19.3].

(6.1) Lemma Let G be a commutative group scheme, flat and of finite type over a noetherian base scheme S. Then G → S is an r.c.i. morphism.

Proof. By definition [EGA IV, 19.3.6], it suffices to check the lemma when S = Spec k, where k is a field. By [EGA IV, 19.3.9(ii)], the property of being an r.c.i. morphism can be checked after an fpqc (faithfully flat and quasi-compact) base change. Therefore, we may and do assume that k is algebraically closed.

Now suppose that

0 → G′ → G → G′′ → 0

is an exact sequence of commutative group schemes over k. Assume that G′ and G′′ are r.c.i. over Spec k. We claim that G → G′′ is an r.c.i. morphism. Indeed, it suffices to check it after the fpqc base change G → G′′, namely, we look at G ×_{G′′} G → G. This morphism is canonically isomorphic to G ×_{Spec k} G′ → G (projection to the first factor), and is an r.c.i. morphism since G′ → Spec k is. Thus the claim is proved. It follows that the composition G → G′′ → Spec k is an r.c.i. morphism, by [EGA IV, 19.3.9(iii)].

Consider an arbitrary G over k. Then G admits a composition series in which the factors are smooth, isomorphic to μ_{p}, or isomorphic to α_{p}. These factors are visibly r.c.i. over k. By the above observation, the lemma is proved.

(6.2) Lemma Suppose that X is a noetherian scheme and X → Spec 𝒪 is a flat r.c.i. morphism. Then for any N ≥ 1, The collection of points:

⋃{x mod π^{N} ∈ X(C^{N}) : x ∈ X(C)}

as C ranges over local 𝒪̂-algebras which are flat, finite, and r.c.i. over 𝒪̂, is schematically dense in X ⊗ 𝒪^{N}.

Proof. Since {Spec 𝒪̂_{X⊗𝒪^{N} ,x} : x ∈ X ⊗ 𝒪^{N}} is obviously schematically dense in X ⊗ 𝒪^{N}, we may assume that X = Spec A is a complete noetherian local ring such that π ∈ 𝔪_{A}.

Choose a presentation A = B/I, where B = 𝒪̂[[X_{1}, …, X_{b}]]. Since the fibers of Spec B → Spec 𝒪̂ are obviously regular, [EGA IV, 19.3.7] implies that Spec A → Spec B is a transversally regular immersion. By definition [EGA IV, 19.2.2], this implies that I is generated by a regular sequence (t_{1}, …, t_{a}) on B. By [EGA0, 15.1.6, (b) implying (c)], (t_{1}, …, t_{a})⊗κ is a regular sequence on B ⊗ κ.

Extend (t_{1}, …, t_{a}) ⊗ κ to a regular sequence on B ⊗ κ of length b = dim B ⊗ κ, and lift the sequence to a sequence (t_{1}, …, t_{b}) in B. Put J_{n} = (t_{1}^{n}, …, t_{b}^{n}). Then ⋂_{n} J_{n} ⊂ ⋂_{n} 𝔪_{B}^{n} = 0. Let C_{n} = B/(I + J_{n}) and Spec C_{n} → X be induced by the obvious map B/I → B/(I + J_{n}). Then the points {Spec C_{n} → X : n ≥ 1} are schematically dense in X.

By [EGA0, 15.1.16, (c) implying (b)], (t_{1}, …, t_{a}, t_{a+1}^{n}, …, t_{b}^{n+1}) is a regular sequence in B and C_{n} = B/(t_{1}, …, t_{a}, t_{a+1}^{n}, …, t_{b}^{n+1}) is flat. Therefore, C_{n} is a flat r.c.i.-R-algebra. It is clearly of relative dimension 0 over 𝒪̂. It follows that C_{n} is finite over 𝒪̂.

It is clear from the above that {Spec C_{n} ⊗ 𝒪^{N} → X ⊗ 𝒪^{N} : n ≥ 1} is schematically dense in X ⊗ 𝒪^{N}.

(6.3) Proposition Let \underline{G} be a commutative noetherian group scheme over 𝒪, not necessarily flat. Let ¯ \underline{G} be the schematic closure of \underline{G} ⊗ K in \underline{G}. Then ¯ \underline{G} ⊗ 𝒪^{N} is the schematic closure in \underline{G} ⊗ 𝒪^{N} of the following collection of points

⋃\{x mod π^{N} ∈ \underline{G}(C^{N}) : x ∈ \underline{G}(C)\}

as C ranges over local 𝒪̂-algebras which are flat, finite, and r.c.i. over 𝒪̂.

Proof. Clear from the two preceding lemmas, together with the fact that ¯ \underline{G}(C) = \underline{G}(C) for any flat 𝒪-algebra C.

(6.4) Lemma The collection of 𝒪^{N}-algebras

{C^{N} : C is a local, flat, finite, r.c.i. 𝒪̂-algebra}

is just the collection of all local 𝒪^{N}-algebras which are flat, finite, and r.c.i. over 𝒪^{N}.

Proof. By [EGA IV, 19.3.9(ii)], the property of being r.c.i. is stable under arbitrary base change. We only need to show that any local, flat, finite, r.c.i. 𝒪^{N}-algebra A is of the form C^{N} for a suitable C.

Choose any presentation A = B/I, B = 𝒪^{N}[X_{1}, …, X_{n}]_{𝔪}, 𝔪 = (π, X_{1}, …, X_{n}). Since the only fiber of Spec B → Spec 𝒪^{N} is obviously regular, [EGA IV, 19.3.7] implies that Spec A → Spec B is a transversally regular immersion. By definition ([EGA IV, 19.2.2]), this implies that the ideal I ⊂ B is generated by a regular sequence (f_{1}, …, f_{m}). We have m = n since A is of relative dimension 0.

Lift f_{i} to f̃_{i} ∈ 𝒪̂[X_{1}, …, X_{n}]𝔪̃, where 𝔪̃ = (π, X_{1}, …, X_{n}). By [EGA 0, 15.1.16, (c) implying (b)],

C = 𝒪̂[X_{1}, …, X_{n}]𝔪̃/(f̃_{1}, …, f̃_{n})

is flat and (f̃_{1}, …, f̃_{n}) is a regular sequence on 𝒪̂[X_{1}, …, X_{n}]𝔪̃. Therefore, C is a local, flat, finite, r.c.i. 𝒪̂-algebra, and A = C^{N} as desired.

§7. Elkik’s theory

(7.1) In this section, Let R be a noetherian 𝒪-algebra, complete with respect to the π-adic topology. Consider R[X] = R[X_{1}, …, X_{N}], the polynomial ring in N variables. Let I be an ideal of R[X] and put B = R[X]/I, Y = Spec B. We assume that Y ⊗_{𝒪} K → Spec(R ⊗_{𝒪} K) is smooth of relative dimension s.

The Jacobian ideal J of I is defined to be the ideal of R[X] generated by the s-minors of

(\frac{∂f_{i}}{∂X_{j}} )_{{1≤i≤s \atop 1≤j≤N}}

for all f_{1}, …, f_{s} in a generating set of I.

Because of our smoothness assumption, J ⊃ π^{h}R[X] for some h ≥ 0 by the Jacobi criterion [BLR, 2.2/Prop. 7]. Fix such an h in the following.

(7.2) Lemma (Elkik) Suppose that I can be generated by N − s elements. Then for any n > 2h, the image of Y (R) → Y (R^{n−h}) is the same as the image of Y (R^{n}) → Y (R^{n−h}).

Proof. This is a special case of [El, p.555, Lemma 1]. Elkik uses an ideal H instead of J as above, and H is defined to be ∑ K_{(α)}Δ_{(α)}, summing over certain tuples (α). Examining the definitions, we see that if we set α = (1, …, s), then K_{(α)} = 1, Δ_{α} = J, and hence H ⊃ J. Therefore the above lemma follows from Elkik’s.

(7.3) Lemma Suppose that R is a local ring, and Y → Spec R is a flat r.c.i. morphism. Then for any n > 2h, the image of Y (R) → Y (R^{n−h}) is the same as the image of Y (R^{n}) → Y (R^{n−h}).

Proof. Let y : Spec R^{n} → Y be a point of Y (R^{n}). Let 𝔪 be the closed point of Spec R^{n}, 𝔮 = y(𝔪).

Since Y → Spec R is r.c.i. at 𝔮 and Spec R[X] → Spec R has regular fibers, f is a transversally regular immersion at 𝔮 ([EGA IV, 19.3.7]). By ([EGA IV, 19.2.2]), there is some f ∈ R[X] such that 𝔮 ∈ Y_{f} ⊂ Spec R[X]_{f} and Y_{f} is cut out by N − s equations on Spec R[X]_{f} = Spec R[X_{1}, …, X_{N}, Z]/(Zf(X) − 1). Because R is local, y maps Spec R^{n} to Y_{f}, i.e. y ∈ Y_{f}(R^{n}).

Let {(g_{1}(X, Z), …, g_{N−s}(X, Z)} be the equations definition Y_{f} on Spec R[X]_{f}. Then we can regard Y_{f} as a closed subscheme of Spec R[X, Z] = 𝔸_{R}^{N+1} defined by the N + 1 − s equations

{g_{1}(X, Z), …, g_{N−s}(X, Z), Zf(X) − 1}.

Let J′ ⊂ R[X, Z] be the Jacobian ideal of these equations. Since J′ is also generated by I and Zf(X) − 1, we see that π^{h} ∈ J′.

By the previous lemma, we can find ỹ ∈ Y_{f}(R) ⊂ Y (R) such that ỹ ≡ y (mod π^{n−h}). This proves the lemma.

§8. The main argument

For simplicity, we assume that K is complete in this section. However, this assumption is not essential. See (8.6) for more comments.

(8.1) The invariant h We will use the group schemes defined in (3.6). Recall from (5.4) that the closed subscheme \underline{T}′ of \underline{R}′ is cut out by (dim R − dim T) equations, and \underline{R}′ = \underline{R} ⊗ 𝒪_{L}.

Since \underline{R} is the Néron model of an induced torus, we can write down its defining equation explicitly. In fact, we can realize \underline{R} as a closed subscheme of 𝔸_{/𝒪}^{d(n+1)}, defined by n explicit equations.

We conclude that \underline{T}′ can be realized as a closed subscheme of 𝔸_{/𝒪_{L}}^{d(n+1)} defined by an ideal I′ generated by d(n + 1) − dim T equations. Let J′ be the Jacobian ideal for I′. Since the generic fiber of \underline{T}′ is smooth, J′ contains π^{h} for some h ≥ 0. Let h = h(𝒪, 𝒪_{L}, Γ, Λ) be the smallest integer with this property.

Lemma Suppose (𝒪, 𝒪_{L}, Γ, Λ) ≡ (𝒪_{0}, 𝒪_{L_{0}}, Γ_{0}, Λ_{0}) (level N). Form the Jacobian ideals J′ and J′_{0} and define the integers h and h_{0} for both data. If h < N or h_{0} < N, then h = h_{0}.

Proof. Assume h < N. The explicit equations generating J′ only depends only on (𝒪, 𝒪_{L}, Γ, Λ), and their reductions modulo π^{N} depends only on (𝒪^{N}, 𝒪_{L}^{N}, 𝒪, 𝒪_{L}). The formation of the Jacobian ideal commutes with base change. Therefore, J′ ⊗ 𝒪^{N} depends only on (𝒪^{N}, 𝒪_{L}^{N}, Γ, Λ). Therefore, our hypothesis implies

J′_{0} + π_{0}^{N}𝒪_{L}[X_{1}, …, X_{d(n+1)}] ⊃ π_{0}^{h}𝒪_{L}[X_{1}, …, X_{d(n+1)}].

Since J′_{0} ⊃ π^{M}𝒪_{L}[X_{1}, …, X_{d(n+1)}] for some M ≫ 0, we have J′_{0} ⊃ π_{0}^{h}𝒪_{L}[X_{1}, …, X_{d(n+1)}] by Nakayama’s lemma. Therefore, h_{0} ≤ h < N. The same argument shows that h ≤ h_{0}. So h = h_{0}.

If h < N, we define h(𝒪^{N}, 𝒪_{L}^{N}, Γ, Λ) to be h, otherwise, we define h(𝒪^{N}, 𝒪_{L}^{N}, Γ, Λ) = N. This definition is justified by the lemma.

(8.2) Proposition The group scheme \underline{T} _{L}^{0} ⊗ 𝒪_{L}^{N−h} is determined by (𝒪^{N}, 𝒪_{L}^{N}, Γ, Λ) if N > 2h.

(8.2.1) Lemma Let C be a local 𝒪_{L}-algebra which is π-adically complete. Then the image of \underline{T}′(C) → \underline{T}′(C^{N−h}) is the same as the image of \underline{T}′(C^{N}) → \underline{T}′(C^{N−h}) if N > 2h.

Proof. By the discussion of (8.1), \underline{T}′ is defined by d(n + 1) − dim T equations on 𝔸^{d(n+1)}, so the result follows from Lemma 7.2.

(8.2.2) We now prove the proposition. First recall (see the end of (3.6)) that \underline{T}′⊗𝒪_{L}^{N} is determined by (𝒪^{N}, 𝒪_{L}^{N}, Γ, Λ).

By (6.3), it suffices to show that the collection of points ⋃_{C} image(\underline{T}′(C) → \underline{T}′(C^{N−h})), where C ranges over all flat, finite, r.c.i. 𝒪̂_{L}-algebras, is determined by (𝒪^{N}, 𝒪_{L}^{N}, Γ, Λ).

By Lemma 8.2.1 and Lemma 6.4, this collection of points is the same as the union of

image(\underline{T}′(C^{N}) → \underline{T}′(C^{N−h}))

over all flat, finite, r.c.i. 𝒪^{N}-algebras C. From this description it is clear that our collection of points is determined by (𝒪^{N}, 𝒪_{L}^{N}, Γ, Λ).

(8.2.3) Remark The argument above shows the following: with the notation of (6.3), assume that \underline{G} ⊗ K is smooth of dimension s, and \underline{G} can be defined by n − s equations on 𝔸_{/𝒪}^{n}, then there is an integer h ≥ 0 such that \underline{G} ⊗ 𝒪^{N} determines ¯ \underline{G} ⊗ 𝒪^{N−h} for all N > 2h.

(8.2.4) Corollary The group scheme \underline{T}^{0} ⊗ 𝒪^{N−h} is determined by (𝒪^{N}, 𝒪_{L}^{N}, Γ, Λ) for N > 2h.

Proof. This follows from the proposition, and the following elementary statement: Suppose X, X′ are closed S-subschemes of an S-scheme Y such that X ×_{S} S′ = X ×_{S} S′ in Y ×_{S} S′ for some S′ → S faithfully flat, then X = X′.

(8.3) In the following, we apply the procedure of (3.4), where R is as in (3.6).

We know (cf. (8.1)) that \underline{T}^{0} is a closed subscheme of 𝔸^{d(n+1)}, defined by an ideal I. We also know that I modulo π^{N−h} depends only on (𝒪^{N}, 𝒪_{L}^{N}, Γ, Λ) if N > 2h.

Let J ⊂ 𝒪[X_{1}, …, X_{d(n+1)}] be the Jacobian ideal of I. Since I′ ⊂ I, we have J′ ⊂ J and π^{h} ∈ J.

(8.4) Proposition i) \underline{T}^{0} ⊗ 𝒪^{N} is determined by (𝒪^{m}, 𝒪_{L}^{m}, Γ, Λ) for all N ≥ 1, m ≥ max(N + h, 2h + 1).

ii) \underline{R}^{i} ⊗ 𝒪^{m−i} depends only on (𝒪^{m}, 𝒪_{L}^{m}, Γ, Λ)) for all m ≥ max(2h + i, 3h + 1).

iii) W^{i} depends only on (𝒪^{m}, 𝒪_{L}^{m}, Γ, Λ) for m ≥ max(2h + i + 1, 3h + 1).

(8.4.1) Lemma If N > 2h, then the image of the map \underline{T}^{0}(𝒪^{sh}) → \underline{T}^{0}((𝒪^{sh})^{N−h}) is the same as the image of \underline{T}^{0}((𝒪^{sh})^{N}) → \underline{T}^{0}((𝒪^{sh})^{N−h}).

Proof. Since \underline{T}^{0} is a flat group scheme over 𝒪, it is r.c.i. over 𝒪 by Lemma 6.1. Now we can apply Lemma 7.3.

(8.4.2) We now prove the proposition. (i) is immediate from Corollary 8.2.4: \underline{T}^{0} ⊗ 𝒪^{N}, is determined by \underline{T}^{0} ⊗ 𝒪^{max(N,h+1)}, hence is determined by (𝒪^{m}, 𝒪_{L}^{m}, Γ, Λ) for all m ≥ max(N, h + 1) + h.

We prove (ii) and (iii) by induction on i. First we look at the case i = 0. (ii) is clear: \underline{R}^{0}(A) = Λ ⊗_{ℤ} ((A ⊗ 𝒪_{L})^{×}) for any 𝒪-algebra A, in particular, for any 𝒪^{N}-algebra A (so actually, in case i = 0, (ii) holds without any restriction on N). (iii): Clearly, W^{0} is determined by the image of \underline{T}^{0}(𝒪^{sh}) → \underline{T}^{0}((𝒪^{sh})^{N}) for any N ≥ 1. In particular, it is determined by the image of \underline{T}^{0}(𝒪^{sh}) → \underline{T}^{0}((𝒪^{sh})^{h+1}). By Lemma 8.4.1, this image is determined by \underline{T}^{0}((𝒪^{sh})^{2h+1}), hence is determined by (𝒪^{m}, 𝒪_{L}^{m}, Γ, Λ) for all m ≥ 3h + 1 by (i).

Now assume that i > 0. (ii) follows immediately from Lemma 4.2 and our induction hypothesis. (iii): Let B^{i} be the affine ring of R^{i}. Recall that

B^{i} = B^{i−1}[Y_{1}, …, Y_{m}]/(πY_{1} − g_{1}, …, πY_{m} − g_{m}) mod π^{∞}-torsion,

where (π, g_{1}, …, g_{m}) is the ideal defining W^{i−1}. We will write suggestively π^{−1}g_{i} for Y_{i}.

A point y on \underline{R}^{i} is determined by the projection of y on \underline{R}^{i−1}, together with the additional “coordinates” (π^{−1}g_{1})(y), …, (π^{−1}g_{m})(y).

Now consider x ∈ \underline{T}^{0}(𝒪^{sh}). Regarding x as an 𝒪^{sh}-valued point on \underline{R}^{i}, we denote it by x_{i}. Clearly, x_{i} mod π is determined by x_{i−1} mod π^{2}. Inductively, the image of \underline{T}^{0}(𝒪^{sh}) → \underline{T}^{0}((𝒪^{sh})^{i+1}) determines W^{i}. As in the case i = 0, we deduce that this image is determined by (𝒪^{m}, 𝒪_{L}^{m}, Γ, Λ) whenever m ≥ max(2h + i + 1, 3h + 1).

The proposition is proved.

(8.5) The main theorem Let

δ = ⌊\frac{δ(e, \underline{T}′)}{e(L/K)}⌋ .

Observe that we have δ ≤ h. If δ < N, we define δ(𝒪^{N}, 𝒪_{L}^{N}, Γ, Λ) to be δ, otherwise, we define δ(𝒪^{N}, 𝒪_{L}^{N}, Γ, Λ) = N. The definition is justified by Lemma (5.5).

Theorem Suppose that N ≥ 1, m ≥ max(N + δ + 2h, 3h + 1), where

\begin{cases}h = h(𝒪^{m}, 𝒪_{L}^{m}, Γ, Λ) & as defined in (8.1), \\ δ = δ(𝒪^{m}, 𝒪_{L}^{m}, Γ, Λ) & as defined above. \end{cases}

Then \underline{T}^{NR} ⊗ 𝒪^{N} is determined by (𝒪^{m}, 𝒪_{L}^{m}, Γ, Λ).

Notice that since δ ≤ h, the hypothesis of the theorem is implied by m ≥ N + 3h.

(8.5.1) Lemma Let X be a smooth scheme over 𝒪. Then the schematic closure of the points

{x : x ∈ X((𝒪^{sh})^{N})} = {x mod π^{N} : x ∈ X(𝒪^{sh})}

in X ⊗ 𝒪^{N} is the whole X ⊗ 𝒪^{N}.

Proof. It suffices to show that if X = Spec A is smooth over 𝒪^{N}, then X((𝒪^{sh})^{N}) is schematically dense in X.

Suppose that f ∈ A is such that x^{∗}f = 0 ∈ (𝒪^{sh})^{N} for all x ∈ X((𝒪^{sh})^{N}). Then f mod π is zero on every closed point of X ⊗ κ^{sep}. Therefore, f ∈ πA. An easy induction shows that f = 0.

(8.5.2) We now prove the theorem. By (3.4), (5.4), and the end of (3.2), \underline{T}^{NR} is the schematic closure of T in \underline{R}^{δ}.

Let Y be the image of \underline{T}^{NR}((𝒪^{sh})^{N}) in \underline{R}^{δ}((𝒪^{sh})^{N}). Then the schematic closure of Y in \underline{R}^{δ} ⊗𝒪^{N} is simply \underline{T}^{NR}⊗𝒪^{N} by the preceding lemma. We know that \underline{R}^{δ}⊗𝒪^{N} is determined by (𝒪^{m}, 𝒪_{L}^{m}, Γ, Λ) by Proposition 8.4(ii). Therefore, it suffices to show that Y is determined by (𝒪^{m}, 𝒪_{L}^{m}, Γ, Λ).

As explained in the proof of Proposition 8.4(iii), Y is determined by the image of

\underline{T}^{0}(𝒪^{sh}) → \underline{T}^{0}((𝒪^{sh})^{δ+N}),

which is determined by the image of \underline{T}^{0}((𝒪^{sh})) → \underline{T}^{0}((𝒪^{sh})^{max(N+δ,h+1)}), which is the same as the image of

\underline{T}^{0}((𝒪^{sh})^{max(N+δ,h+1)+h}) → \underline{T}^{0}((𝒪^{sh})^{max(δ+N,h+1)})

by Lemma 8.4.1.

By Corollary 8.2.4, \underline{T}^{0}((𝒪^{sh})^{max(δ+N,h+1)+h}) is determined by (𝒪^{m}, 𝒪_{L}^{m}, Γ, Λ). The theorem has been proved completely.

(8.6) Remark For the simplicity of exposition, we have assumed that K is complete. This is because we want to apply Elkik’s result in Lemma 8.4.1. We can lift this restriction by either using the henselian version of Elkik’s result, or by observing that both \underline{T}^{NR} ⊗ 𝒪^{N} and (𝒪^{m}, 𝒪_{L}^{m}, Γ, Λ) remain unchanged when we replace K by its completion.

§9. Application of Deligne’s theory

(9.1) Deligne’s theory We recall some notions and results from Deligne’s article [D]. In this section, assume that K is complete and the residue field κ is perfect.

Let e ≥ 1. A Galois extension L/K is at most e-ramified if Gal(L/K)^{e} = {1}, where the superscript e refers to the upper numbering filtration by the ramification groups [S1]. In other words, Gal(L/K) is a quotient of Gal(K^{sep}/K)/ Gal(K^{sep}/K)^{e}. Deligne [D] shows that

Gal(K^{sep}/K)/ Gal(K^{sep}/K)^{e},

together with its upper numbering filtration, is canonically determined by Tr_{e} K, which is defined in the introduction. Therefore, we may denote Gal(K^{sep}/K)/ Gal(K^{sep}/K)^{e} by Γ(Tr_{e} K).

Suppose that Tr_{e} K is isomorphic to Tr_{e} K_{0} and L/K is a Galois extension, at most e-ramified. Then there is a corresponding L_{0}/K_{0}. This can be explicated as follows ([D, 1.3]). Suppose that φ : 𝒪^{e} → 𝒪_{0}^{e}, η : 𝔭/𝔭^{e+1} → 𝔭_{0}/𝔭_{0}^{e+1} defines the isomorphism Tr_{e} K → Tr_{e} K_{0}. Let π_{L} be a prime element of 𝒪_{L} satisfying the Eisenstein equation

X^{n} + ∑_{i=1}^{n−1} a^{(i)}X^{i} = 0

with a^{(i)} ∈ 𝔭. Let a_{0}^{(i)} ∈ 𝒪_{0} be a life of η(a^{(i)} mod 𝔭^{e+1}). Then the equation X^{n} + ∑_{i=1}^{n−1} a_{0}^{(i)}X^{i} = 0 defines the extension L_{0}/K_{0}.

It is clear from the above description that the isomorphism Tr_{e} K → Tr_{e} K_{0} determines canonically an isomorphism 𝒪_{L}/π^{e} → 𝒪_{L_{0}}/π_{0}^{e}, which is Γ(Tr_{e} K)-equivariant.

(9.2) Reformulation of the main theorem Now let S be a triple isomorphic to Tr_{e} K, and Λ be a ℤ-lattice on which Γ(S) acts continuously. Let Γ be the (finite) image of Γ(S) in Aut(Λ). Let L/K be the extension whose Galois group is Γ. By the preceding discussion, (S, Λ) determines (𝒪^{e}, 𝒪_{L}^{e}, Γ, Λ) canonically.

We define h(S, Λ) to be h(𝒪^{e}, 𝒪_{L}^{e}, Γ, Λ). Now we have the following reformulation of Theorem 8.5:

Theorem Let S = (R, M, ϵ) be a triple as in [D, 1.1], and let Λ be a lattice on which Γ(S) acts continuously. Suppose that N ≥ 1, e ≥ N + 3h(S, Λ). Then there exists canonically a smooth group scheme 𝒯_{N}(S, Λ) defined over R/𝔪_{R}^{N} with the following property: for any complete local field K and for any isomorphism (φ, η) : S → Tr_{e} K, regard Λ as a Gal(K^{sep}/K)-module via Deligne’s theory and let T_{/K} be the associated torus, then there is a canonical isomorphism

𝒯_{N}(S, Λ) ⊗_{φ} 𝒪^{N} ≃ \underline{T}^{NR} ⊗ 𝒪^{N}.

(9.3) Maintain the preceding notations. As in (3.6), we denote the Néron model of T ⊗ L by \underline{T}^{†}. We have trivially that the diagram in (3.6) modulo π^{m} is determined by (S, Λ).

The closed immersion T_{L}^{0} ⊗ 𝒪_{L}^{N} → \underline{T}^{†} ⊗ 𝒪_{L}^{N} is determined by (S, Λ). This is clear from the proof of (8.2). From our main argument, it is clear that the morphisms \underline{R}^{i+1} ⊗ 𝒪^{N} → \underline{R}^{i} ⊗ 𝒪^{N}, as well as the closed immersion \underline{T}^{NR} ⊗ 𝒪^{N} → \underline{R}^{δ} ⊗ 𝒪^{N}, are determined by (S, Λ).

It follows that the morphism \underline{T}^{NR} ⊗ 𝒪^{N} → \underline{R}^{0} ⊗ 𝒪^{N} is determined by (S, Λ). This morphism factors through the closed immersion \underline{T}^{0} ⊗ 𝒪^{N} → \underline{R}^{0} ⊗ 𝒪^{N}. It follows that \underline{T}^{NR} ⊗ 𝒪^{N} → \underline{T}^{0} ⊗ 𝒪^{N} is determined by (S, Λ).

Finally, we conclude that the morphism

\underline{T}^{NR} ⊗ 𝒪_{L}^{N} → \underline{T}^{†} ⊗ 𝒪_{L}^{N}

is determined by (S, Λ).

(9.4) The construction Λ ↦ 𝒯_{N}(S, Λ) is functorial: Let Λ_{1}, Λ_{2} be two lattices on which Γ(S) acts continuously and let λ : Λ_{1} → Λ_{2} is a homomorphism of Γ(S)-modules, then for e ≫ N there is a canonical morphism 𝒯_{N}(S, λ) : 𝒯_{N}(S, Λ_{1}) → 𝒯_{N}(S, Λ_{2}) which corresponds to the reduction modulo π^{N} of the canonical morphism λ_{∗} : \underline{T} 1^{NR} → \underline{T} 2^{NR} induced by λ.

We sketch the proof: let L/K be a Galois extension splitting both T_{1} and T_{2}. Let

h_{i} = h(𝒪^{m}, 𝒪_{L}^{m}, Γ, Λ_{1}), i = 1, 2.

Assume that m ≥ N + 4 max(h_{1}, h_{2}).

From the preceding paragraph, we know that the morphism \underline{T} i^{NR} ⊗ 𝒪^{N+h} → \underline{R}i^{NR} ⊗ 𝒪^{N+h} is determined by (𝒪^{m}, 𝒪_{L}^{m}, Γ, Λ_{i}). Given a point point x ∈ \underline{T} i^{NR}((𝒪^{sh})^{N}), lift x to x̃ ∈ \underline{T} i^{NR}((𝒪^{sh})^{N+h}), then the image of x̃ in \underline{R}i^{NR}((𝒪^{sh})^{N+h}) determines x by the proof of Prop. 8.4(iii).

Consider the commutative diagram:

\underline{R}1^{NR}((𝒪^{sh})^{N+h}) ⟶ \underline{R}2^{NR}((𝒪^{sh})^{N+h})
↑  ↑
\underline{T} 1^{NR}((𝒪^{sh})^{N+h}) ⟶^{?} \underline{T} 2^{NR}((𝒪^{sh})^{N+h})
↓  ↓
\underline{T} 1^{NR}((𝒪^{sh})^{N}) ⟶^{?} \underline{T} 2^{NR}((𝒪^{sh})^{N})

The whole diagram, except possibly the two arrows labelled with question marks, is determined by (𝒪^{m}, 𝒪_{L}^{m}, Γ, Λ_{1}, Λ_{2}). Given x ∈ \underline{T} 1^{NR}((𝒪^{sh})^{N}), we lift x to x̃ ∈ \underline{T} 1^{NR}((𝒪^{sh})^{N+h}). Let y be the image of x̃ under the composition \underline{T} 1^{NR}((𝒪^{sh})^{N+h}) → \underline{R}1^{NR}((𝒪^{sh})^{N+h}) → \underline{R}2^{NR}((𝒪^{sh})^{N+h}). Then y is also the image of x̃ under the composition \underline{T} 1^{NR}((𝒪^{sh})^{N+h}) → \underline{T} 2^{NR}((𝒪^{sh})^{N+h}) → \underline{R}2^{NR}((𝒪^{sh})^{N+h}). It follows that y determines λ_{∗}(x) ∈ \underline{T} 2^{NR}((𝒪^{sh})^{N}).

From the schematic density of \underline{T} 1^{NR}((𝒪^{sh})^{N}) in \underline{T} 1^{NR} ⊗ 𝒪^{N}, it follows that the morphism

\underline{T} 1^{NR} ⊗ 𝒪^{N} → \underline{T} 2^{NR} ⊗ 𝒪^{N}

is determined by (𝒪^{m}, 𝒪_{L}^{m}, Γ, Λ_{1}, Λ_{2}).

§10. The invariant c(T)

(10.1) As in the introduction, we define an invariant of a torus T over K as follows: by the universal property of the Néron model, there is a canonical morphism

can_{T,L} : \underline{T}^{NR} ⊗ 𝒪_{L} → \underline{(T ⊗ L)}^{NR}

extending the identity morphism on the generic fibers. Then can_{T,L} induces an inclusion on the Lie algebras. We define

c(T) = \frac{1}{e(L/K)} length_{𝒪_{L}} \frac{Lie(\underline{(T ⊗ L)}^{NR})}{(can_{T,L})_{∗}(Lie \underline{T}^{NR} ⊗ 𝒪_{L})}.

It is easy to see that this is independent of the splitting field L.

The module of translation invariant 1-forms on \underline{T} /𝒪^{NR} (resp. T_{/K}) is a free 𝒪-module (resp. Kvector space) of rank n = dim T (see [BLR, 4.2]), and is canonically isomorphic to Hom_{𝒪} (Lie \underline{T}^{NR}, 𝒪) (resp. Hom_{K}(Lie T, K)). The top exterior power of this module can be identified with the space of translation invariant n-forms on \underline{T} /𝒪^{NR} (resp. T_{/K}), and is denoted by ω(\underline{T}^{NR}) (resp. ω(T)).

Then we can also define c(T) by

c(T) = \frac{1}{e(L/K)} length_{𝒪_{L}} \frac{ω(\underline{T}^{NR}) ⊗ 𝒪_{L}}{can_{T,L}^{∗} ω(\underline{(T ⊗ L)}^{NR})}

(10.2) Proposition The following two assertions are equivalent:

i) c(T_{1}) = c(T_{2}) for any two tori T_{1}, T_{2} over K such that T_{1} is isogenous to T_{2} over K. ii) c(T) = \frac{1}{2}a(X_{∗}(T)⊗ℚ) for any torus T over K, where a(−) is the Artin conductor of a module

over ℚ[Gal(K^{sep}/K)].

Proof. This is essentially in [GG]. We briefly sketch their argument. Clearly, we have (ii) implying (i). By a direct calculation, we can show that c(T) = \frac{1}{2}a(X_{∗}(T) ⊗ ℚ) when T = Res_{L/K} 𝔾_{m} is an induced tori.

Assume (i). Then the function X_{∗}(T) ↦ c(T) extends to an additive function on R ⊗ ℚ with values in ℚ, where R is the ring of virtual representations over ℚ of Gal(K^{sep}/K). A theorem of Artin [S2, 9.2] implies that R ⊗ ℚ is generated by X_{∗}(T) ⊗ ℚ as T runs through tori of the form Res_{L/K} 𝔾_{m}. Now the proposition is obvious.

(10.3) The rest of this section gives an interesting condition which is equivalent to the assertion that c(T) is invariant under isogeny.

The space ω(T ⊗L) is canonically isomorphic to X^{∗}(T)⊗L, where X^{∗}(T) is the character group of T.

Now suppose that we have two tori T_{1} and T_{2} over K and an isogeny α : T_{1} → T_{2} defined over K corresponding to α^{∗} : X^{∗}(T_{2}) → X^{∗}(T_{1}). Then the isomorphism

α^{♯} = \frac{1}{deg f} ⋀^{top}(α^{∗}) : ⋀^{top} X^{∗}(T_{2}) → ⋀^{top} X^{∗}(T_{1})

induces an isomorphism

(α ⊗ L)^{♯} : ω(T_{2} ⊗ L) → ω(T_{1} ⊗ L).

It is easy to show ([GG]) that (α ⊗ L)^{♯} descends to an isomorphism α^{♯} : ω(T_{2}) → ω(T_{1}). We call α^{♯} the modified-pullback. When K is of characteristic 0, it is simply the ordinary pull-back divided by deg α.

(10.4) Lemma With above notations. c(T_{1}) = c(T_{2}) if and only if α^{♯} : ω(T_{2}) → ω(T_{1}) induces an isomorphism

α^{♯} : ω(\underline{T} 2^{NR}) → ω(\underline{T} 1^{NR}).

Proof. We have a commutative diagram

ω(T_{1}) ⊗ L ⟶^{can_{T_{1},L}^{∗}} ω(T_{1} ⊗ L)
↑_{α^{♯}⊗L}  ↑^{(α⊗L)^{♯}}
ω(T_{2}) ⊗ L ⟶^{can_{T_{2},L}^{∗}} ω(T_{2} ⊗ L)

in which each arrow is an isomorphism and each object X is a 1-dimensional vector space over L containing a canonical 𝒪_{L}-lattice M_{X}. For each arrow g : X → Y, we let c(g) = length M_{Y} /g(M_{X}) if M_{Y} ⊃ g(M_{X}), c(g) = − length g(M_{X})/M_{Y} otherwise. Clearly, c(g ∘ h) = c(g) + c(h).

It is clear that c((α ⊗ L)^{♯}) = 0. Therefore, c(α^{♯} ⊗ L) = 0 if and only if c(can_{T_{1},L}) = c(can_{T_{2},L}). By definition, c(α^{♯} ⊗ L) = 0 if and only if α^{♯}(ω(\underline{T} 2^{NR})) = ω(\underline{T} 1^{NR}), and c(can_{T_{i},L}^{∗}) = e(L/K) · c(T_{i}), i = 1, 2. The lemma follows.

§11. Isogeny invariance in characteristic 0

(11.1) It would be useful to prove that the invariant c(T) does not change under isogeny when the local field K has characteristic 0. When the residue field also has characteristic zero, this is not difficult: There is only tame ramification, and the desired result follows from an application of [Ed, Thm. 4.2].

On the other hand when the residue field of 𝒪_{K} has characteristic p > 0, the isogeny invariance can be proved using Galois cohomology. We offer two proofs. The first one is an arithmetic proof, using Tate’s Euler-Poincaré characteristic formula, and is valid when the residue field is finite. The second proof is more general and more geometric; but the idea is the same.

The lemma below may be of some interest. It may be convenient but not strictly necessary for the proof of isogeny invariance.

(11.2) Lemma Let K be a field equipped with a discrete valuation and let T be a torus over K Let T_{s} be the maximal split subtorus of T, and let T_{a} be the quotient torus T/T_{s}. Then the canonical arrows

1 ⟶ \underline{T} s^{NR} ⟶ \underline{T}^{NR} ⟶ \underline{T} a^{NR} ⟶ 1

form a short exact sequence.

Proof. Writing T_{s} as a successive extension of 𝔾_{m}’s, we may and do assume that T′ ≅ 𝔾_{m}. As in the proof of [BLR, 7.5/Prop. 1], it suffices to show that the extension

1 ⟶ T_{s} ⟶ T ⟶ T_{a} ⟶ 1

extends to an exact sequence of smooth group schemes

1 ⟶ \underline{T} s^{NR} ⟶ \underline{T}^{∗} ⟶ \underline{T} a^{NR} ⟶ 1

for some smooth model \underline{T}^{∗} of T. But this is exactly the statement of [SGA7I, VIII Cor. 6.6].

(11.3) For the rest of this section, K denotes a local field with mixed characteristic (0, p) and perfect residue field. The theorem below was stated in [G, Prop. 4.7] when the residue field of K is finite.

Theorem Let T_{1}, T_{2} be two tori over K, and let α : T_{1} → T_{2} be a K-isogeny. Then the two tori have the same invariant:

c(T_{1}) = c(T_{2}) = \frac{1}{2}a(X_{∗}(T) ⊗ ℚ)

(11.4) Proposition Consider the pull-back map α^{∗} : ω(\underline{T} 2^{NR}) → ω(\underline{T} 1^{NR}) as above. There exists an element a ∈ 𝒪_{K}, unique up to 𝒪_{K}^{×}, such that 𝒪_{K} · α^{∗}(ω(\underline{T} 2^{NR})) = a · ω(\underline{T} 1^{NR}). Denote the rational number p^{ord_{p}(a)} by deg_{diff}(α), which we call the differential degree of α. Then

deg_{diff}(α) ≤ p^{ord_{p}(deg α)}

In the above, ord_{p} denotes the valuation on K with ord_{p}(p) = 1.

(11.4.1) Proof of Proposition 11.4 when K is a finite extension of ℚ_{p}.

By Lemma 11.2, we may and do assume that T_{1} and T_{2} are anisotropic over the maximal unramified extension of K (replacing K by a finite unramified extension if necessary). Then \underline{T} i^{NR}(L) = T_{i}(L) for any unramified extension L/K, i = 1, 2.

Let \underline{T} 1^{NR}^{∘} (resp. \underline{T} 2^{NR}^{∘}) be the neutral component of the Néron model \underline{T} 1^{NR} (resp. \underline{T} 2^{NR}). Let ω_{i} be an 𝒪_{K}-generator of ω(\underline{T} i^{NR}), i = 1, 2. Let n = ord_{K}(a) = ord_{K}(deg_{diff}(α)). Let M = ker(α), the kernel of the isogeny α.

Consider finite unramified extensions L of K. Let q_{L} be the cardinality of the residue field κ_{L} of 𝒪_{L}. Let |ω_{2}| (resp. |α^{∗}ω_{2}|) be the Haar measure on \underline{T} 2^{NR}(𝒪_{L}) (resp. \underline{T} 1^{NR}(𝒪_{L})) attached to ω_{2} (resp. α^{∗}ω_{2}). Considering the volumes of \underline{T} 2^{NR}^{∘}(𝒪_{L}) \underline{T} 1^{NR}^{∘}(𝒪_{L}), we have

|α^{∗}ω_{2}|(\underline{T} 1^{NR}^{∘}(𝒪_{L})) = Card(M(L) ∩ \underline{T} 1^{NR}^{∘}(𝒪_{L})) · |ω_{2}|(α(\underline{T} 1^{NR}^{∘}(𝒪_{L})))

By definition, for i = 1, 2, |ω_{i}|(\underline{T} i^{NR}(𝒪_{L})) is equal to the number of q_{L}-rational points of the closed fibre of \underline{T} i^{NR}^{∘}, divided by q_{L}^{dim T_{i}}. Since T_{i} is anisotropic, its closed fibre is a unipotent group over κ_{L} and has the same number of κ_{L}-rational points as 𝔸^{dim(T_{i})}. Hence |ω_{1}|(\underline{T} 1^{NR}(𝒪_{L})) = |ω_{2}|(\underline{T} 2^{NR}(𝒪_{L})), and the previous formula gives

[\underline{T} 2^{NR}^{∘}(𝒪_{L}) : α(\underline{T} 1^{NR}^{∘}(𝒪_{L}))] = Card(M(L) ∩ \underline{T} 1^{NR}^{∘}(𝒪_{L})) · q^{n}

Notice that Card(M(L) ∩ \underline{T} 1^{NR}^{∘}(𝒪_{L})) become constant if L is large enough.

On the other hand, let C_{T_{1}} (resp. C_{T_{2}}) be the group of geometric connected components of the closed fibre of \underline{T} 1^{NR} (resp. \underline{T} 2^{NR}). For sufficiently large finite unramified extensions L of K, we have

[\underline{T} 2^{NR}^{∘}(𝒪_{L}) : α(\underline{T} 1^{NR}^{∘}(𝒪_{L}))] = \frac{Card(C_{T_{1}})}{Card(C_{T_{2}})} · [\underline{T} 2^{NR}(𝒪_{L}) : α(\underline{T} 1^{NR}(𝒪_{L}))]

Combined with the previous paragraph, we see that the differential degree deg_{diff}(α) of α can be detected from the growth behavior of

[T_{2}(L) : α(T_{1}(L))] = [\underline{T} 2^{NR}(𝒪_{L}) : α(\underline{T} 1^{NR}(𝒪_{L}))]

as L runs through finite unramified extensions of K.

On the other hand, by Tate’s formula for the Euler-Poincaré characteristic for the Galois cohomologies of local fields (see [S3, II Thm. 5]) we have

Card(H^{1}(L, M)) = q_{L}^{ord_{K}(deg α)} · Card(M(L)) · Card(H^{2}(L, M))

By the local duality for Galois cohomology of local fields ([M, I Cor. 2.3]), H^{2}(L, M) is the dual of M^{D}(L), where M^{D} is the Cartier dual of the finite group scheme M over K. This finite group M^{D}(L) become constant if L is sufficiently large.

From the long exact sequence of Galois cohomologies attached to the isogeny α, we get an injection from T_{2}(L)/α(T_{1}(L)) to H^{1}(L, M). This gives us the required bound

deg_{diff}(α) ≤ p^{ord_{p}(deg α)}

(11.4.2) Proof of Proposition 11.4, the general case.

We may and do assume that K is complete and the residue field κ of K is algebraically closed. As before, we may and do assume that T_{1} and T_{2} are anisotropic. Our proof is similar to [R, p.208–210] in flavor.

For i = 1, 2 and for each n > 0, let \underline{T}_{i,n} be the Greeberg functor applied to \underline{T} i^{NR} ×_{Spec 𝒪} Spec 𝒪^{n}. Each \underline{T}_{i,n} is a smooth group scheme over the residue field κ, of dimension nd = n dim(T_{i}). We have canonical isomorphisms \underline{T}_{i,n}(κ) = \underline{T}_{i}(𝒪^{n}). Let

\underline{T}_{i} = \underleftarrow{lim}_{n} \underline{T}_{i,n}

where the projective system is regarded as a projective system in the category of quasi-algebraic groups, and the limit is a pro-algebraic group in the sense of Serre [S4]. We have canonical isomorphisms

\underline{T}_{i}(κ) = \underline{T} i^{NR}(𝒪_{K}) = T_{i}(K)

The isogeny α induces homomorphisms \underline{α}_{n} : \underline{T}_{1,n} → \underline{T}_{2,n} and \underline{α} : \underline{T}_{1} → \underline{T}_{2}. Let \underline{M} (resp. \underline{C}) be the kernel (resp. cokernel) of \underline{α}. We have an exact sequence

0 → \underline{M} → \underline{T}_{1} → \underline{T}_{2} → \underline{C} → 0

Let M be the kernal of α, a finite group scheme over K. The kernel \underline{M} is a finite quasi-algebraic group over κ. We claim that the dimension of the quasi-algebraic group \underline{C} is equal to deg_{diff}(α).

To prove this claim, we proceed as in [R, 2.2.1]. Choose N ≫ 0 such that for i = 1, 2, the exponential map induces an isomorphism from π^{N} Lie \underline{T} i^{NR} to ker(\underline{T} i^{NR}(𝒪) → \underline{T} i^{NR}(𝒪^{N})). The group π^{N} Lie \underline{T} i^{NR} has an obvious structure of pro-algebraic group \underline{π^{N} Lie \underline{T} i^{NR}} over κ. Recall from [S4, 2.4, Prop. 7] that the category of pro-algebraic groups is abelian. The following map between short exact sequences of pro-algebraic groups

0 ⟶ \underline{π^{N} Lie \underline{T} 1^{NR}} ⟶ \underline{T}_{1} ⟶ \underline{T}_{1,N} ⟶ 0
\underline{Lie α}↓  ↓\underline{α}  ↓\underline{α}_{N}
0 ⟶ \underline{π^{N} Lie \underline{T} 2^{NR}} ⟶ \underline{T}_{2} ⟶ \underline{T}_{2,N} ⟶ 0

gives rise to an exact sequence

0 → \underline{M} → ker(\underline{α}_{N}) → coker(\underline{Lie α}) → \underline{C} → coker(\underline{α_{N}}) → 0

We deduce from the exact sequence that dim \underline{C} = dim coker(\underline{Lie α}), which is equal to the length of the cokernel of π^{N}Lie(\underline{T} 1^{NR}) → π^{N}Lie(\underline{T} 2^{NR}). This proves the claim.

By [Be, 4.3.3], the cohomology group H^{1}(K, M) has a natural structure as a quasi-algebraic group \underline{H^{1}(K, M)} over κ of dimension ord_{p}(deg α). The long exact sequence attached to

0 → M → T_{1} → T_{2} → 0

gives us an injection from \underline{C} into \underline{H^{1}(K, M)}. Hence dim \underline{C} ≤ dim \underline{H^{1}(K, M)}. The proposition is proved.

(11.5) Proof of Theorem 11.3.

By Prop. 10.2, it suffices to show c(T_{1}) = c(T_{2}). This follows easily from the bound provided by Prop 11.4: Choose an isogeny β : T_{2} → T_{1} such that β ∘ α = [n]_{T_{1}} for some integer n > 0. Write n = p^{a} · u, where a = ord_{p}(n), u ∈ ℤ, and let d = dim T_{1} = dim T_{2}. We have

p^{ad} = deg_{diff}(β ∘ α) = deg_{diff}(β) deg_{diff}(α) ≤ p^{ord_{p}(deg α)}p^{ord_{p}(deg β)} = p^{ad}

So the equality holds throughout the above displayed inequality. This implies that c(T_{1}) = c(T_{2}) and the theorem is proved.

§12. Isogeny invariance in characteristic p

(12.1) Theorem Assume that K is of equal-characteristic p and the residue field of 𝒪_{K} is perfect. Let T be a torus over K. Then

c(T) = \frac{1}{2}a(X_{∗}(T) ⊗ ℚ).

In particular, it is invariant under isogeny.

Proof. Without loss of generalities, we may replace K by its completion. So we now assume that K is complete.

From the definition, it is clear that the invariant c(T) is determined by the morphism

\underline{T}^{NR} ⊗ 𝒪_{L}^{m} → \underline{(T ⊗ L)}^{NR} ⊗ 𝒪_{L}^{m}

for m ≫ 0. By the discussion of (9.3), it follows that c(T) is determined by (Tr_{e} K, Λ) for e ≫ 0.

Choose e large enough such that c(T) is determined by (Tr_{e} K, Λ). Choose a local field K_{0} of characteristic 0 such that Tr_{e} K_{0} ≃ Tr_{e} K. By applying Theorem 11.3 to the torus over K_{0} determined by (Tr_{e} K_{0}, Λ), we see that c(T) = c(T_{0}) is one-half of the Artin conductor of Λ as a Gal(K_{0}^{sep}/K_{0})-module. This Artin conductor is determined by Λ as a module over Γ(Tr_{e} K_{0}) = Γ(Tr_{e} K) and the upper numbering filtration on Γ(Tr_{e} K_{0}) = Γ(Tr_{e} K), and is equal to the Artin conductor of Λ as a Gal(K^{sep}/K)-module. The theorem is proved.

(12.2) Remark Let 0 ≤ c_{1} ≤ ⋯ ≤ c_{n} be the exponents of the elementary divisors of the following torsion module

\frac{Lie \underline{(T ⊗ L)}^{NR}}{(can_{T,L})_{∗}(Lie \underline{T}^{NR}) ⊗ 𝒪_{L}} ,

divided by e(L/K). Then each c_{i} = c_{i}(T) is an invariant of T_{/_{K}} and c(T) = ∑ c_{i}(T). Our main theorem shows that c_{i}(T) depends only on (T_{e}(K), Λ) for e ≫ 0. But we don’t know anything else about these invariants.

Appendix : Another approach to the characteristic p case

by Ehud de Shalit

§A1. A recipe for the Lie algebra

In the appendix we describe a second approach to Theorem 12.1. It is also based on the Deligne-Kazhdan-Krasner idea that local fields in characteristic p are limits of local fields in characteristic 0, as the absolute index of ramification grows to infinity. But it deals directly with the Lie algebra, rather than the Néron model. In other words, if the main idea behind the first approach was to show that the reduction of the Néron model modulo π^{N} depended functorially only on some “truncated data”, we do the same here with the Lie algebra. Although the result is weaker, it is sufficient to prove Theorem 12.1 (which motivated this work), and the proof is somewhat simpler.

As before, let K be a local field, T a torus over K, and L a finite Galois extension of K splitting T. Let Γ = Gal(L/K). We proceed in three steps. In the first, outlined in the present section, we obtain a recipe for the Lie algebra of the Néron model of T in terms of the character group M = X^{∗}(T) of the torus, as a Γ-module. This recipe is of independent interest, and does not seem to follow from the previous sections. In the second step, we extend some old results of S. Sen [Sen] on the Galois cohomology of local fields in characteristic 0. Then, in Prop. A3.2, we show that the invariant c(T) depends on “truncated data” only. Our Sen-type result implies that in characteristic 0, the depth of the truncation depends on M, the Galois group Γ, and its ramification filtration, but not on the actual local field K. This “uniformity” result in Galois cohomology is essential in the last step, where we apply Deligne’s method to reduce the main theorem in characteristic p, to the characteristic 0 case (as in Theorem 11.3 above).

It is interesting to note that in addition to Deligne’s theory, there are other points of contact between the two approaches. For example, the use of Elkik’s result in the first approach, finds an analogue in the second, both in our use of M. Artin’s approximation theorem in Lemma A1.6 below, and later on, when we make precise the depth of the truncation on which c(T) depends.

The appendix is logically independent of §3–§12, except for admitting Theorem 11.3 (the main theorem in characteristic 0) as an input. We tried to make it notationally self-contained too, although cross-references are inevitable.

(A1.1) Generalities on tori As in §2, we consider a local field K, a separable closure K^{sep}, and a finite Galois extension L of K, contained in K^{sep}. Then Γ = Gal(L/K) is a quotient of G = Gal(K^{sep}/K). We let G × Γ act on Γ on the left via (g × γ)(σ) = gσγ^{−1} (g ∈ G, γ, σ ∈ Γ). We let G × Γ act on K^{sep} via the projection to G. We then have a canonical ring-isomorphism, respecting the action of G × Γ,

K^{sep} ⊗_{K} L ≃ Maps(Γ, K^{sep})  (A1.1)

sending a ⊗ b to the map σ ↦ a · σ(b).

Let T be a torus over K, split by L, and

M = X^{∗}(T) = Hom(T_{/K^{sep}}, 𝔾_{m/K^{sep}})  (A1.2)

the character group. As a Γ-module, M is the dual lattice of Λ = X_{∗}(T). As functors on K-algebras we have the canonical identification

T (−) = Hom_{Γ}(M, (− ⊗_{K} L)^{×}).

Indeed, the right hand side defines a torus T_{1} whose character group is (M ⊗ ℤΓ)_{Γ} (coinvariants) and the map m ⊗ σ ↦ σm is an isomorphism (M ⊗ ℤΓ)_{Γ} ≃ M carrying the G action on the first to the original Γ action on the second. Therefore T_{1} is the original T.

Let T′ = T ×_{Spec K} Spec L. Then T′(−) = Hom(M, (−)^{×}) is the canonical trivialization of T over L.

(A1.2) Néron models We need a watered-down version of the results explained in detail in Section 3. Let S = Spec(𝒪_{K}) and S′ = Spec(𝒪_{L}). Let 𝒩 (resp. 𝒩′) be the Néron (lft-)model of T (resp. T′) over S (resp. S′) and let 𝒯 = 𝒩^{0} (resp. 𝒯′ = 𝒩^{′0}) be its neutral component.

(A1.3) Proposition ([BLR, 10.1/4]) 𝒩 is the group smoothening of the schematic closure of T in the Weil restriction of scalars Res_{S′/S} 𝒩′.

See [BLR, p.174] for the notion of group smoothening. It is defined by a universal property, but in practice is carried out by successive dilatations as described in §4, each time reducing Néron’s defect of smoothness, until we reach a smooth group scheme.

Since T′ is split, we have, as functors on 𝒪_{L}- and 𝒪_{K}-algebras, the identities

𝒯′(−) = Hom(M, (−)^{×}), Res_{S′/S} 𝒯′(−) = Hom(M, (− ⊗_{𝒪_{K}} 𝒪_{L})^{×}).  (A1.3)

We shall make use of the intermediate model 𝒳, defined as a functor on 𝒪_{K}-algebras by

𝒳(−) = Hom_{Γ}(M, (− ⊗_{𝒪_{K}} 𝒪_{L})^{×}),  (A1.4)

and we let 𝒴 stand for the schematic closure of T in 𝒳. Of course, 𝒴_{K} = 𝒳_{K} = T, and 𝒴 is a closed subscheme of 𝒳.

In concrete terms, Res_{S′/S} 𝒯′ = Spec A, T = Spec A_{K}/I and the ideal I is generated, in A_{K}, by a collection {f_{i}} of polynomials expressing the condition that a homomorphism from M to (− ⊗_{𝒪_{K}} 𝒪_{L})^{×} be Γ-invariant. Introducing a ℤ-basis in M, it is clear that these polynomials are in fact in A. Then 𝒳 = Spec A/{f_{i}}, and 𝒴 = Spec A/ℐ, where ℐ = ker(A → A_{K}/I). The {f_{i}} may not generate ℐ over A, because there could be an A_{K}-linear combination of them lying in A, which is not an A-linear combination. However, the following lemma means that for our purposes the difference between 𝒴 and 𝒳 is unimportant.

(A1.4) Lemma For any 𝒪_{K}-algebra R which embeds in R_{K}, 𝒴(R) = 𝒳(R).

Proof. Letting A, I and ℐ be as above, take x ∈ 𝒳(R), and let x_{K} be its generic fiber. Then we have a diagram

[figure: I; A_{K}; x_{K}; R_{K}; ℐ; A; x; R]  (A1.5)

and x_{K}(I) = 0. It follows that x(ℐ) = 0, and x factors through 𝒴.

Since 𝒳 is a closed subscheme of Res_{S′/S} 𝒯′ = Res_{S′/S}(𝒩^{′0}), 𝒴 is the schematic closure of T in Res_{S′/S}(𝒩^{′0}), so (by Prop. A1.3) 𝒯 is the neutral component of the group smoothening of 𝒴. (Note that even though the special fiber of 𝒴 is connected, the special fiber of its group smoothening need not be connected.)

(A1.5) Lie algebras If R = 𝒪_{K}[[t]] or R = 𝒪_{K}[ε] (ε = t mod t^{2}, ε^{2} = 0), we let, for any S-group scheme 𝒢

𝒢(R)_{1} = ker (𝒢(R) → 𝒢(𝒪_{K})) .  (A1.6)

We also write R_{1} for 𝔾_{m}(R)_{1} = ker(R^{×} → 𝒪_{K}^{×}). Clearly 𝒢(R)_{1} ⊂ 𝒢^{0}(R). If 𝒢 is smooth over S, then by definition

Lie 𝒢 = 𝒢(𝒪_{K}[ε])_{1}.  (A1.7)

Consider the diagram

[figure: 𝒯(O_{K}[[t]])_{1}; ũ; 𝒴(𝒪_{K}[[t]])_{1}; Hom_{Γ}(M, 𝒪_{L}[[t]]_{1}); Lie 𝒯 = 𝒯(𝒪_{K}[ε])_{1}; u; 𝒴(𝒪_{K}[ε])_{1}; Hom_{Γ}(M, 𝒪_{L}).]  (A1.8)

We make the following observations regarding the diagram.

• The left vertical arrow is surjective because 𝒯 is smooth ([BLR], 2.2/6).

• The maps u and ũ are the group smoothening maps.

• The equality signs on the right are based on the last lemma. In the bottom right corner we identified 𝒪_{L}[ε]_{1} ≃ 𝒪_{L}ε ≃ 𝒪_{L}.

• The map u is injective because both Lie 𝒯 and Hom_{Γ}(M, 𝒪_{L}) are 𝒪_{K}-submodules of Lie T = Hom_{Γ}(M, L).

(A1.6) Lemma The map ũ is an isomorphism.

Proof. The injectivity of ũ is proved as that of u, and in fact is not needed below. To show that ũ is surjective, let A_{0} = 𝒪_{K}[t]_{(π,t)}, a local ring. Then

A_{0} ⊂ A = A_{0}^{h} ⊂ Â = 𝒪_{K}[[t]],  (A1.9)

where A is the henselization of A_{0}, and Â its completion. By M. Artin’s approximation theorem ([BLR], 3.6/16) any ŷ ∈ 𝒴(Â) can be approximated in the (π, t)-adic topology arbitrarily well by y ∈ 𝒴(A). But A is the direct limit of smooth 𝒪_{K}-algebras, and for any smooth 𝒪_{K}-algebra R, 𝒯(R) = 𝒴(R) ([BLR], 7.1/5). Thus y lifts to a point of 𝒯(A), and we deduce that Im(ũ) is dense. As 𝒪_{K}[[t]] is compact and ũ is continuous in the (π, t)-adic topology, Im(ũ) is also closed, so ũ is surjective.

The discussion above leads immediately to our “recipe” for the Lie algebra of T.

(A1.7) Proposition (i) Canonically and functorially

Lie(𝒯) = {v ∈ Hom_{Γ}(M, 𝒪_{L}) | v lifts to Hom_{Γ}(M, 𝒪_{L}[[t]]_{1})}  (A1.10)

as a lattice in Lie(T) = Hom_{Γ}(M, L).

(ii) By the universal property of 𝒩′ there is a map 𝒯 ×_{S} S′ → 𝒯′. The associated map on tangent spaces is

[figure: Lie(𝒯) ⊗_{𝒪_{K}} 𝒪_{L}; Lie(𝒯′); Hom_{Γ}(M, 𝒪_{L}) ⊗_{𝒪_{K}} 𝒪_{L}; Hom(M, 𝒪_{L}).]  (A1.11)

(A1.8) Example: Restriction of scalars Consider T = Res_{L/K} 𝔾_{m}. Then M = ℤΓ. It follows that Hom_{Γ}(M, 𝒪_{L}) = 𝒪_{L}, and every v lifts, so the inclusion Lie(𝒯)⊗_{𝒪_{K}} 𝒪_{L} ↪ Lie(𝒯′) is the inclusion 𝒪_{L} ⊗_{𝒪_{K}} 𝒪_{L} ↪ Hom(ℤΓ, 𝒪_{L}) sending a⊗b to the homomorphism σ ↦ σ(a)·b (σ ∈ Γ). Let δ_{σ} be the homomorphism which is 1 on σ and 0 on τ ∈ Γ, τ ≠ σ, and let {α} be a basis of 𝒪_{L} over 𝒪_{K}. Then in terms of the 𝒪_{L}-bases {α ⊗ 1} and {δ_{σ}} the matrix of the inclusion 𝒪_{L} ⊗_{𝒪_{K}} 𝒪_{L} ↪ Hom(ℤΓ, 𝒪_{L}) is given by (σ(α)) . Its determinant, up to a unit, is \sqrt{Δ_{L/K}}, the square root of the discriminant. Let a(M_{ℚ}) be the (exponent of the) Artin conductor of the rational representation M_{ℚ}. Then by the conductor-discriminant formula a(M_{ℚ}) = a(ℚΓ) = ord_{K}(Δ_{L/K}), where ord_{K} is the normalized valuation on K. It follows that in this case the invariant c(T) defined in §10.1 is equal to a(M_{ℚ})/2. This verifies the main theorem for tori which are restriction of scalars, or products of such.

(A1.9) The invariant c(T) It is instructive to compute more examples using Prop. A1.7, and note which “part” of the invariant c(T) is attributed to the failure to lift every v ∈ Hom_{Γ}(M, 𝒪_{L}) to Hom_{Γ}(M, 𝒪_{L}[[t]]_{1}). For lack of space we do not do it here. In general, we may write

c(T) = c_{0}(T) + c_{lift}(T)  (A1.12)

where

c_{0}(T) = e(L/K)^{−1} length_{𝒪_{L}} (Hom(M, 𝒪_{L})/ Hom_{Γ}(M, 𝒪_{L}) ⊗_{𝒪_{K}} 𝒪_{L}) ,  (A1.13)

and

c_{lift}(T) = e(L/K)^{−1} length_{𝒪_{L}} (Hom_{Γ}(M, 𝒪_{L}) ⊗_{𝒪_{K}} 𝒪_{L}/ Lie(𝒯) ⊗_{𝒪_{K}} 𝒪_{L})
= length_{𝒪_{K}} (Hom_{Γ}(M, 𝒪_{L})/ Lie(𝒯)) .  (A1.14)

To analyze c_{lift}(T) introduce the following notation. For any commutative ring k and an additive subgroup I ⊂ k write R(k) = k[[t]] and

\begin{cases}R_{i}(k) = 1 + t^{i}R(k) ⊂ R(k)^{×}, \\ R_{i}(I) = 1 + t^{i}I[[t]], \end{cases}  i ≥ 1.  (A1.15)

The short exact sequence of Γ-modules

0 → R_{2}(𝒪_{L}) → R_{1}(𝒪_{L}) → 𝒪_{L} → 0,  (A1.16)

yields, upon applying Hom(M, −) and taking the long exact sequence in cohomology, an exact sequence

0 → Lie(𝒯) → Hom_{Γ}(M, 𝒪_{L}) → H^{1}(Γ, Hom(M, R_{2}(𝒪_{L}))).  (A1.17)

We have taken here into account the characterization of Lie(𝒯) given by Prop. A1.7.

§A2. Some results in Galois cohomology

Before we proceed we need to develop some old ideas of S.Sen about the Galois cohomology of local fields in characteristic 0 [Sen]. In this section, let E/F be a finite Galois extension of local fields in characteristic 0. Thus F (and E) is a finite extension of ℚ_{p}. Let π_{F} (resp. π_{E}) be a uniformizer of F (resp. E). The group Γ = Gal(E/F) is equipped with its (lower and upper) ramification filtrations, {Γ_{u}} and {Γ^{v}}. The upper and lower filtrations determine each other, by means of the Herbrand functions (see [S1]). Let M be a Γ-module, free of finite rank over ℤ. As in §2.4, we shall say that a quantity “X” depends on (M, Γ, fil) if there is a canonical way to compute “X” from knowledge of Γ, the Γ-module M, and the ramification filtrations on Γ. The emphasis here will be on independence of the actual field extension E/F. For example, the exponent a(M_{ℚ}) of the Artin conductor depends only on (M, Γ, fil) by its very definition. Note that this is not true for the Artin conductor itself. The more ramified F is, the smaller the Artin conductor becomes.

(A2.1) Lemma There is an integer ν, depending only on (M, Γ, fil), such that

H^{1}(Γ, Hom(M, 𝒪_{E}))

is killed by π_{F}^{ν} .

Proof. (i) When M = ℤ and Γ is cyclic of prime order, this is Theorem 2 of [Sen].

(ii) When M = ℤ but Γ is arbitrary, we reduce immediately to the case where Γ is a p-group. But then Γ is nilpotent, so this case follows from (i) by induction on |Γ|, using the inflation-restriction exact sequence. (Note that Theorem 3 of [Sen] is not strong enough for our purposes.)

(iii) Let M and Γ be arbitrary. Write W = Hom(M, 𝒪_{E}). Let T be the torus over F, split by E, whose character group is M. By Theorem 11.3, which we assume valid in the characteristic 0 case,

e(E/F)^{−1} length_{𝒪_{E}} (W/W^{Γ} ⊗_{𝒪_{F}} 𝒪_{E}) = c_{0}(T) ≤ c(T) = a(M_{ℚ})/2  (A2.1)

so W/W^{Γ} ⊗_{𝒪_{F}} 𝒪_{E} is killed by a power of π_{F} depending on (M, Γ, fil) only, and the same is true for H^{1}(Γ, W/W^{Γ} ⊗_{𝒪_{F}} 𝒪_{E}). But the sequence

H^{1}(Γ, W^{Γ} ⊗_{𝒪_{F}} 𝒪_{E}) → H^{1}(Γ, W) → H^{1}(Γ, W/W^{Γ} ⊗_{𝒪_{F}} 𝒪_{E})  (A2.2)

is exact. The first term is equal to W^{Γ} ⊗_{𝒪_{F}} H^{1}(Γ, 𝒪_{E}), which is killed by a power of π_{F} depending only on (M, Γ, fil) by case (ii). It was already observed that the same holds for the last term. Hence H^{1}(Γ, W) as well is killed by a power of π_{F} depending only on (M, Γ, fil) and the lemma is proved.

(A2.2) Lemma There is an integer N, depending only on (M, Γ, fil), such that

H^{1}(Γ, Hom(M, R_{i}(𝒪_{E}))) ↪ H^{1}(Γ, Hom(M, R_{i}(𝒪_{E}/π_{F}^{N}𝒪_{E})))  (A2.3)

is injective for every i ≥ 1.

Proof. Clearly, if the lemma holds with one N, it holds with any larger N as well. Let ν be as in the previous lemma, and N ≥ 2ν. Suppose that {φ_{σ}} is a 1-cocycle in Hom(M, R_{i}(𝒪_{E})), which becomes a 1-coboundary in Hom(M, R_{i}(𝒪_{E}/π_{F}^{N}𝒪_{E})). Modifying it, we may assume to begin with that φ_{σ} ∈ Hom(M, R_{i}(π_{F}^{N}𝒪_{E})). If

φ_{σ}(m) = 1 + a_{σ}(m)t^{i} + ⋯ (m ∈ M)  (A2.4)

then {a_{σ}} is a 1-cocycle in Hom(M, π_{F}^{N}𝒪_{E}). It follows from the previous lemma that there exists a b ∈ Hom(M, π_{F}^{N−ν}𝒪_{E}) such that a_{σ} = σ(b) − b (i.e. a_{σ}(m) = σ(b(σ^{−1}m)) − b(m)).

Pick a basis m_{1}, …, m_{d} of M, and define β ∈ Hom(M, R_{i}(π_{F}^{N−ν}𝒪_{E})) by

β(∑_{j=1}^{d} k_{j}m_{j}) = ∏_{j=1}^{d}(1 + b(m_{j})t^{i})^{k_{j}}.  (A2.5)

Then

• β(m) = 1 + b(m)t^{i} + ⋯

• β(m) ≡ 1 + b(m)t^{i} (mod π_{F}^{N}𝒪_{E}).

The second point follows from N ≥ 2ν. Now (σ(β)/β)(m) = 1 + a_{σ}(m)t^{i} + ⋯ ≡ 1 mod π_{F}^{N}𝒪_{E}. Modifying the cocycle {φ_{σ}} by the coboundary {σβ/β} we see that the same cohomology class is represented by a cocycle in Hom(M, R_{i+1}(π_{F}^{N}𝒪_{E})). Continuing in this way we see that the cohomology class of {φ_{σ}} is trivial, and the map in the lemma is therefore injective.

If φ ∈ Hom_{Γ}(M, 𝒪_{E}) we write φ^{N} for its image in Hom_{Γ}(M, 𝒪_{E}/π_{F}^{N}𝒪_{E}).

(A2.3) Corollary Let L/K be a finite Galois exetension of local fields, Γ = Gal(L/K), and M a Γ-module, free of finite rank over ℤ. Then there exists an N such that φ ∈ Hom_{Γ}(M, 𝒪_{L}) lifts to Hom_{Γ}(M, R_{1}(𝒪_{L})) if and only if φ^{N} ∈ Hom_{Γ}(M, 𝒪_{L}/π_{K}^{N}𝒪_{L}) lifts to Hom_{Γ}(M, R_{1}(𝒪_{L}/π_{K}^{N}𝒪_{L})). Furthermore, in the characteristic 0 case this N depends only on (M, Γ, fil).

Proof. Let Φ be the subgroup of φ ∈ Hom_{Γ}(M, 𝒪_{L}) which lift, Φ^{N} the subgroup of those φ for which φ^{N} lifts to Hom_{Γ}(M, R_{1}(𝒪_{L}/π_{K}^{N}𝒪_{L})), and Φ̄^{N} the subgroup of Hom_{Γ}(M, 𝒪_{L}/π_{K}^{N}𝒪_{L}) which lift to Hom_{Γ}(M, R_{1}(𝒪_{L}/π_{K}^{N}𝒪_{L})). From the discussion in §A1.9, and from the analogous discussion modulo π_{K}^{N} we obtain the commutative diagram

\xymatrix{0 \ar[r] & Φ \ar[r] \ar[d] & Hom_{Γ}(M, 𝒪_{L}) \ar[r] \ar[d] & H^{1}(Γ, Hom(M, R_{2}(𝒪_{L}))) \ar[d] \\ 0 \ar[r] & Φ̄^{N} \ar[r] & Hom_{Γ}(M, 𝒪_{L}/π_{K}^{N}𝒪_{L}) \ar[r] & H^{1}(Γ, Hom(M, R_{2}(𝒪_{L}/π_{K}^{N}𝒪_{L}))),} (A2.6)

in which the rows are exact. In the characteristic 0 case, the last lemma shows that for N large enough, depending only on (M, Γ, fil), the last vertical arrow is injective. Thus φ ∈ Φ if and only if φ^{N} ∈ Φ̄^{N}.

In general, Proposition A1.7 identifies Φ = Lie 𝒯 and it follows that Φ is of finite index in Hom_{Γ}(M, 𝒪_{L}), because both are 𝒪_{K}-lattices of the same rank. Since

Φ ⊂ Φ^{N+1} ⊂ Φ^{N} ⊂ Hom_{Γ}(M, 𝒪_{L}),  (A2.7)

the sequence of subgroups Φ^{N} stabilizes for large N. But it is clear, by compactness, that if φ is such that φ^{N} lifts for every N, then φ itself lifts. Therefore Φ = Φ^{N} for large enough N, and the claim is proved.

§A3. Reduction of the char. p case to the char. 0 case

(A3.1) The invariant c(T) depends only on truncated data Let L/K, Γ and M be as in the last corollary, and let T be the torus over K, split by L, whose character group is M. Let 𝒯 be as before, the neutral component of the Néron model of T. Choose m large enough such that the following conditions hold.

M1 π_{K}^{m} annihilates Hom(M, 𝒪_{L})/ Lie(𝒯) ⊗_{𝒪_{K}} 𝒪_{L},

M2 If char. K = 0, π_{K}^{m} kills H^{1}(Γ, Hom(M, 𝒪_{L})). If char. K = p, we require only that the map

H^{1}(Γ, Hom(M, 𝒪_{L}))[π_{K}^{m}] ↪ H^{1}(Γ, Hom(M, 𝒪_{L}))/π_{K}^{m}H^{1}(Γ, Hom(M, 𝒪_{L}))  (A3.1)

is injective. Note that for any finitely generated 𝒪_{K}-module Λ, Λ[π_{K}^{m}] ↪ Λ/π_{K}^{m}Λ is injective for large enough m.

M3 An element φ of Hom_{Γ}(M, 𝒪_{L}) lifts to Hom_{Γ}(M, R_{1}(𝒪_{L})) if and only if φ^{m} lifts to Hom_{Γ}(M, R_{1}(𝒪_{L}/π_{K}^{m}𝒪_{L})).

By Theorem 11.3, Lemma A2.2 and the last corollary, if char. K = 0, then how large m has to be depends only on (M, Γ, fil).

(A3.2) Proposition The invariant c(T) depends only on (M, Γ, fil) and 𝒪_{L}/π_{K}^{2m}𝒪_{L}, viewed as a Γ-module. More precisely, let there be given two Galois extensions L/K and E/F of local fields, and an isomorphism Gal(L/K) ≃ Gal(E/F) preserving the ramification filtrations. Call this common group Γ and endow it with the resulting filtrations. Let M be a Γ-module, free of finite rank over ℤ. Let T_{K} and T_{F} be the corresponding tori. Let m be an integer for which the above conditions hold, both for L/K and for E/F. Suppose that there is an isomorphism of Γ-modules

𝒪_{L}/π_{K}^{2m}𝒪_{L} ≃ 𝒪_{E}/π_{F}^{2m}𝒪_{E}.  (A3.2)

Then c(T_{K}) = c(T_{F}).

Proof. Consider the commutative diagram with exact rows

[figure: 0; 𝒪_{L}; π_{K}^{m}; 𝒪_{L}; can; 𝒪_{L}/π_{K}^{m}; 0; 0; 𝒪_{L}/π_{K}^{m}; π_{K}^{m}; 𝒪_{L}/π_{K}^{2m}; can; 𝒪_{L}/π_{K}^{m}; 0.]  (A3.3)

Its top row yields a short exact sequence

0 → Hom_{Γ}(M, 𝒪_{L})/π_{K}^{m} → Hom_{Γ}(M, 𝒪_{L}/π_{K}^{m}) → H^{1}(Γ, Hom(M, 𝒪_{L}))[π_{K}^{m}] → 0,  (A3.4)

and also an injection

H^{1}(Γ, Hom(M, 𝒪_{L}))/π_{K}^{m} ↪ H^{1}(Γ, Hom(M, 𝒪_{L}/π_{K}^{m})).  (A3.5)

By M2 we find a short exact sequence

0 → Hom_{Γ}(M, 𝒪_{L})/π_{K}^{m} → Hom_{Γ}(M, 𝒪_{L}/π_{K}^{m}) ⟶^{δ_{m}} H^{1}(Γ, Hom(M, 𝒪_{L}/π_{K}^{m})).  (A3.6)

The second row of the diagram shows that δ_{m} is computed from knowledge of 𝒪_{L}/π_{K}^{2m} alone. Thus

Hom_{Γ}(M, 𝒪_{L})/π_{K}^{m} = ker(δ_{m}),  (A3.7)

as a subgroup of Hom_{Γ}(M, 𝒪_{L}/π_{K}^{m}), depends on 𝒪_{L}/π_{K}^{2m} only.

Let Φ_{m} be the image of Lie(𝒯) in ker(δ_{m}). By M1, Lie(𝒯) ⊃ π_{K}^{m} Hom_{Γ}(M, 𝒪_{L}), so

[Hom_{Γ}(M, 𝒪_{L}) : Lie(𝒯)] = [ker(δ_{m}) : Φ_{m}].  (A3.8)

By M3, Φ_{m} = ker(δ_{m}) ∩ ker(ε_{m}), where

ε_{m} : Hom_{Γ}(M, 𝒪_{L}/π_{K}^{m}) → H^{1}(Γ, Hom(M, R_{2}(𝒪_{L}/π_{K}^{m})))  (A3.9)

is the connecting homomorphism as in the second row of (A2.6). We conclude that c_{lift}(T), which is computed from the index

[ker(δ_{m}) : ker(δ_{m}) ∩ ker(ε_{m})],  (A3.10)

depends only on 𝒪_{L}/π_{K}^{2m}.

Next, to compute c_{0}(T) we need to know the index

[Hom(M, 𝒪_{L}) : Hom_{Γ}(M, 𝒪_{L}) ⊗_{𝒪_{K}} 𝒪_{L}],  (A3.11)

which, by M1, is equal to

[Hom(M, 𝒪_{L}/π_{K}^{m}) : ker(δ_{m}) ⊗_{𝒪_{K}/π_{K}^{m}} 𝒪_{L}/π_{K}^{m}].  (A3.12)

This last formula shows that it, too, depends only on 𝒪_{L}/π_{K}^{2m}.

(A3.3) Conclusion of the proof We are now ready to apply the Deligne-Kazhdan-Krasner idea. We use, as in §9, Deligne’s formalism. Recall that we assume the validity of Theorem 11.3, and want to derive Theorem 12.1.

Fix a local field K, char. K = p, a torus T = T_{K} over K, and let L/K be a Galois splitting field. Then Γ = Gal(L/K) is equipped with its ramification filtrations. Let M = X^{∗}(T) be the character group of T, a ℤ-lattice with a Γ-action.

Choose m large enough such that the three conditions on m at the beginning of this section hold for the particular extension L/K, and also for any extension of local fields E/F in characteristic 0, for which

(Gal(E/F), fil) ≃ (Γ, fil).  (A3.13)

This is possible by the remark preceding the proposition.

Let e be large enough so that Γ^{e} = 1. Let F be any local field of characteristic 0 such that, in Deligne’s category of truncated valuation rings,

Tr_{e}(F) ≃ Tr_{e}(K).  (A3.14)

(See [D], or the introduction, for the notion of “truncation”). Such an F always exists. One simply has to take the absolute index of ramification of F large enough. According to the main theorem of Deligne ([D], Th. 2.8), fixing such an isomorphism induces an equivalence of categories

(ext F)^{e} ≈ (ext Tr_{e}(F))^{e} ≈ (ext Tr_{e}(K))^{e} ≈ (ext K)^{e},  (A3.15)

between the category of at-most e-ramified extensions of F and the same category for K. In particular there exists then a unique Galois extension E/F corresponding to L/K, and an isomorphism preserving the ramification filtrations,

Gal(E/F) ≃ Γ = Gal(L/K).  (A3.16)

Let e(1) be computed from e as in [D] “Construction 3.4.1”. Note that it depends only on (Γ, fil) and e. If e is large enough so that

π_{F}^{2m}|π_{E}^{e(1)}  (A3.17)

then there is a Γ-isomorphism

𝒪_{E}/π_{F}^{2m}𝒪_{E} ≃ 𝒪_{L}/π_{K}^{2m}𝒪_{L}  (A3.18)

(loc.cit.).

Let T_{F} be the torus over F, split by E, whose character group is M, viewed as a Gal(E/F)- module via the above identification of the Galois group with Γ. Prop. A3.2 now, finally, implies

c(T_{K}) = c(T_{F}) = a(M_{ℚ})/2,  (A3.19)

and the proof of Theorem 12.1 is concluded.

References

[Be] L. Bégueri: Dualité sur un corps local à corps résiduel algébriquement clos, Mémoire Soc. Math. de France, supplément 108, 121p. (1980)

[BLR] S. Bosch, W. Lütkebohmert, and M. Raynaud: Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete 21, Springer Verlag (1990)

[D] P. Deligne: Les corps locaux de caractéristique p, limites de corps locaux de caractéristique 0, in: Bernstein et al.: Représentations des groupes réductifs sur un corps local, 119–157, Hermann, Paris (1984)

[dS] E. de Shalit: Artin-Schreier-Witt extensions as limits of Kummer-Lubin-Tate extensions, and the explicit reciprocity law, in: D. Goss et al. eds, The arithmetic of function fields, de Gruyter, New York (1992)

[Ed] B. Edixhoven: Néron models and tame ramification, Compositio Math. 81, 291–306 (1992)

[EGA] A. Grothendieck and J. Dieudonné: Eléments de Géométrie Algébrique, Chapire 0, §14– §23, Publ. Math. I.H.E.S. 20 (1964), Chapitre IV, §16–§19, Publ. Math. I.H.E.S. 32 (1967)

[El] R. Elkik: Solutions d’équations a coefficients dans un anneau hensélien, Ann. Scient. Éc. Norm. Sup. 6, 553–604 (1973)

[G] B.H. Gross: On the motive of a reductive group, Inv. Math. 130, 287–313 (1997)

[GG] B.H. Gross and W.T. Gan: Haar measures and the Artin conductor, Trans. Amer. Math. Soc. 351, no. 4, 1691–1704 (1999)

[K] E. Kushnirsky, Ph.D. Thesis, Univ. of Michigan (1999)

[M] J.S. Milne: Arithmetic duality theorems, Perspectives in Mathematics 1, Academic Press (1986)

[P] G. Prasad: Volumes of S-arithmetic quotients of semisimple groups, Publ. Math. I.H.E.S. 69, 91–117 (1989)

[R] M. Raynaud: Hauteurs et isogénies, Astérisque 137, 199–234 (1985)

[S1] J.-P. Serre: Local fields, Springer-Verlag (1979)

[S2] J.-P. Serre: Linear representations of finite groups, Springer-Verlag (1977)

[S3] J-P. Serre: Cohomologie Galoisienne, Lecture Notes in Math. vol. 5, Springer-Verlag (1964)

[S4] J.-P. Serre: Groupes proalgébriques, Publ. Math. I.H.E.S. 7, 4–68 (1960)

[Sen] S. Sen: On automorphisms of local fields, Ann. Math. 90 33-46 (1969)

[SGA7I] A. Grothendieck: Groupes de Monodromie en Géométrie Algébrique, (SGA 7) I, Lecture Notes in Math. vol. 288, Springer-Verlag, (1972)

Ching-Li Chai  Jiu-Kang Yu
Department of Mathematics  Department of Mathematics
University of Pennsylvania  Princeton University
Philadelphia, PA 19104y  Princeton, NJ 08540
U.S.A.  U.S.A.

Email: chai@math.upenn.edu  Email: yu@math.princeton.edu

Ehud de Shalit

Institute of Mathematics

Hebrew University

Giv’at-Ram, Jerusalem 91904

ISRAEL

               email: deshalit@math.huji.ac.il