Elementary divisors of the base change conductor for tori

Ching-Li Chai^{1}

Version 1.0, July 3, 2001

§1. Introduction

Let K be a local field, and let 𝒪 be the ring of integers of K. In this paper we study some numerical invariants attached to tori over K, or equivalently, to integral Galois representations Gal(K^{sep}/K) → GL_{d}(ℤ) with finite image.

Let T be a torus over K. The definition of the numerical invariants

\underline{c}(T, K) = (c_{1}(T, K), …, c_{dim(T)}(T, K))

of T can be found in 2.4. The c_{i}(T, K)’s are non-negative rational numbers satisfying

c_{1}(T, K) ≤ ⋯ ≤ c_{dim(T)}(T, K) .

They come from the elementary divisors of the map from the Lie algebra of the Néron model \underline{T}^{NR} to the Lie algebra of the Néron model \underline{T_{L}}^{NR} over 𝒪, where L is a finite Galois extension of K such that the torus T is split over L. One may think of \underline{(}T, K) as a measure for the failure of T to have semistable reduction over 𝒪: A torus T extends to a torus over 𝒪 if and only if c_{i}(T, K) = 0 for i = 1, …, dim(T). We call the c_{i}(T, K)’s the elementary divisors of the base change conductors; their sum c(T, K) := c_{1}(T, K) + ⋯ + c_{dim(T)}(T, K) is called the base change conductor of T.

According to [CYdS, 11.3, 12.1] or [dS], c(T, K) is equal to one-half of the Artin conductor of the representation of the Galois group Gal(K^{sep}/K) on the character group (or the cocharacter group) of T, if the residue field κ of 𝒪 is perfect. In particular c(T, K) = c(T_{1}, K) if T and T_{1} are isogenous over K. In contrast, the c_{i}(T, K)’s tend to change under isogenies; see 5.2 for an example.

Using the correspondence between a torus T and its character group X_{∗}(T), one can regard \underline{c}(T, K) as a numerical invariant \underline{c}(ρ) attached to a continuous Galois representation ρ : Gal(K^{sep}/K) → GL_{d}(ℤ). Assume that Char(κ) = p > 0. It is easy to see that if ρ_{1} and ρ_{2} are two Galois representations on ℤ^{d} which are conjugate in GL_{d}(ℤ_{(p)}), then \underline{c}(ρ_{1}) = \underline{c}(ρ_{2}). So one can localize at p and define \underline{c}(ρ) for any continuous Galois representation ρ : Gal(K^{sep}/K) → GL_{d}(ℤ_{(p)}). In the next paragraph we describe some general estimates of these invariants in terms of the ramification of the Galois representation ρ.

Let r ≥ 0 be the integer such that c_{r}(T, K) = 0, c_{r+1}(T, K) > 0, where c_{0}(T, K) = 0 by convention, and T is assumed to be non-trivial. Assuming that the residue field κ is perfect, and T is split over a finite Galois extension L of K. Let Γ = Gal(L/K) be the Galois group of L/K, and let Γ = Γ_{−1} ⊇ Γ_{0} ⊇ ⋯ be the lower numbering filtration of Γ. Denote by t be the integer such that t ≥ −1 and Γ = Γ_{t} ⊋ Γ_{t+1}. The results of this paper gives the following estimate:

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

\frac{t + 1}{e(L/K)} ≤ c_{r+1}(T, K) ≤ c_{dim T} (T, K) ≤ \frac{ord_{K}(disc(L/K))}{e(L/K)} (†)

See Cor. 4.3 and Cor. 4.5. In a sense these bounds cannot be improved: Denote by R_{L/M} := Res_{L/K}(𝔾_{m}) the Weil restriction of 𝔾_{m} from L to K. If L/K is totally ramified, then r = 1, dim(R_{L/M}) = [L : K], c_{2}(R_{L/M}, K) = \frac{t+1}{e(L/K)}, and c_{[L:K]}(R_{L/M}, K) = \frac{ord_{K}(disc(L/K))}{e(L/K)}. On the other hand in the context of (†), we do not have any non-trivial general estimate of c_{i}(T, K) if i is in the “intermediate range” r + 1 ≤ i ≤ dim(T) − 1.

Here is a description the method used in this paper. In Thm. 3.1 we prove a series of inequalities relating \underline{c}(T, K) to \underline{c}(T′, K) and \underline{c}(T′′, K), whenever there is an exact sequence of tori 1 → T′ → T → T′′ → 1. In Thm. 4.1 we compute all the c_{i}(R_{L/K}, K)’s for the Weil restriction R_{L/K}, where L is a totally ramified separable extension of K. In Thm. 4.4 we compute the first invariant c_{1}(R′_{L/K}, K) for the norm-one subtorus

R′_{L/K} := ker (Nm_{L/K} : R_{L/K} → 𝔾_{m})

in R_{L/K}(𝔾_{m}), where L/K is assumed to be Galois and totally ramified. The estimate (†) follows from these computation and inequalities provided by 3.1.

We would like to stress that the results in this paper about the invariants c_{i}(T, K) is just a small step, under the benevolent assumption that the residue field is perfect. We have not even computed the invariants c_{i}(R′_{L/K}, K), i ≥ 2 for the norm-one torus R′_{L/K}. For a continuous Galois representation ρ : Gal(K^{sep}/K) → GL(ℤ_{(p)}), we do not know to what extent \underline{c}(ρ) is related to \underline{c}(ρ′), where ρ′ is the contragredient representation of ρ. If the residue field κ is not perfect, then the base change conductor c(T, K) may change under K-isogenies; see 5.3 for such an example. It remains a challenge to determine which estimates about the invariants c_{i}(T, K) proved under the assumption that κ is perfect remain true without that assumption.

§2. Notation and definitions

(2.1) In this paper 𝒪 denotes a complete discrete valuation ring with fraction field K. Let κ be the residue field of 𝒪, 𝔭 be the maximal ideal of 𝒪, and let π be a generator of 𝔭. For a finite separable extension field L of K, denote by 𝒪_{L}, κ_{L}, 𝔭_{L}, π_{L} the ring of integers of L, the residue field of 𝒪_{L}, the maximal ideal of 𝒪_{L}, and a generator of π_{L} respectively. The ramification index e(L/K) is defined by π 𝒪_{L} = π_{L}^{e(L/K)}. Let 𝒪^{sh} be the strict henselization of 𝒪_{L}. The fraction field K^{sh} of 𝒪^{sh} is the maximal unramified extension of K, and the residue field of 𝒪^{sh} is κ^{sep}, the separable closure of κ. We assume that the characteristic of the residue field κ is a prime number p. No generality is sacrificed by this assumption: Every finite extension of a local field is at most tamely ramified if the residue field has characteristic 0, in which case the questions treated in this paper are vastly simplified.

(2.2) Néron models Let T be a torus over K. We have two notions of Néron models of T. The lft Néron model \underline{T}^{lft NR} as defined in [BLR, Ch. 10], is a smooth group model of T locally of finite type over 𝒪, and satisfies \underline{T}^{lft NR}(𝒪^{sh}) = T (K^{sh}). We also have the open subgroup \underline{T}^{ft NR} of \underline{T}^{lft NR} such that \underline{T}^{ft NR}(𝒪^{sh}) is the maximal bounded subgroup of T (K^{sh}); \underline{T}^{lft NR} is of finite type over 𝒪. We abbreviate \underline{T}^{ft NR} to \underline{T}^{NR}.

(2.3) Definition Let 𝒪_{L} be a discrete valuation ring with fraction field L. Let N_{1} ⊂ N_{2} be two free 𝒪_{L}-modules of finite rank d. Let h : N_{1} → N_{2} be an injective 𝒪_{L}-homomorphism. Let j be a natural number such that 0 ≤ j ≤ d. Define α_{j}(h; N_{1}, N_{2}) to be the largest natural number m such that the Λ^{j}(h) ≡ 0 (mod π_{L}^{m}), where Λ^{j}(h) : Λ_{𝒪_{L}}^{j} (N_{1}) → Λ_{𝒪_{L}}^{j} (N_{2}) is the jth exterior power of h. In other words, if π_{L}^{e_{1}} ≤ π_{L}^{e_{2}} … ≤ π_{L}^{e_{dim(V )}} are the elementary divisors of the 𝒪_{L}-module homomorphism h, then α_{j}(h; N_{1}, N_{2}) = e_{1} + ⋯ + e_{j}. When N_{1} ⊆ N_{2} are two lattices in an d-dimensional vector space and h : N_{1} → N_{2} is the inclusion map, then we abbreviate α_{j}(h; N_{1}, N_{2}) to α_{j}(N_{1}, N_{2}). For any integer n ≥ 0, α_{1}(h; N_{1}, N_{2}) ≥ n means that h ≡ 0 (mod π_{L}^{n}), while α_{d}(h; N_{1}, N_{d}) − α_{d−1}(h; N_{1}, N_{2}) ≤ n means that coker(h) is killed by π_{L}^{n}.

(2.4) Definition Let T be a torus over K, and let L be a finite Galois extension of K such that T is split over L. Let

can_{T,L/K} : \underline{T}^{NR} ×_{Spec 𝒪} Spec 𝒪_{L} → \underline{T_{L}}^{NR}

be the unique homomorphism which extends the identity map between the generic fibers, where T_{L} := T ×_{Spec K} Spec L. Let (can_{T,L/K})_{∗} : Lie(\underline{T}^{NR}) ⊗_{𝒪} 𝒪_{L} → Lie(\underline{T_{L}}^{NR}) be the homomorphism between the Lie algebras induced by can_{T,L/K}. Define

\underline{c}(T, K) = (c_{1}(T, K), …, c_{dim(T)}(T, K)) ∈ ℚ_{≥0}^{d}

by

c_{1}(T, K) + ⋯ + c_{i}(T, K) = \frac{1}{e(L/K)} α_{i} ((can_{T,L/K})_{∗}; Lie(\underline{T}^{NR}) ⊗_{𝒪} 𝒪_{L}, Lie(\underline{T_{L}}^{NR})) ,

for i = 1, …, dim(T). In other words, c_{1}(T, K) ≤ ⋯ ≤ c_{dim(T)}(T, K), and

π_{L}^{e(L/K)c_{1}(T,K)}, …, π_{L}^{e(L/K)c_{dim(T)}(T,K)}

are the elementary divisors of the 𝒪_{L}-module homomorphism (can_{T,L/K})_{∗}. The above definition first appeared in [CYdS, 12.2]; it does note depend on the choice of L. The sum c(T, K) := c_{1}(T, K) + ⋯ + c_{dim(T)}(T, K), called the base change conductor of T in [Ch1], was studied in [CYdS]. We call the c_{i}(T, K)’s the elementary divisors of th base change conductor of T.

(2.5) Let L be a finite separable extension of K. Denote by R_{L/K} the torus Res_{L/K}(𝔾_{m}), such R_{L/K}(A) = (A ⊗_{K} L)^{×} for every K-algebra A. Its Néron model \underline{R_{L/K}}^{NR} is equal to Res_{𝒪_{L}/𝒪} (𝔾_{m}), the Weil restriction of 𝔾_{m} from 𝒪_{L} to 𝒪.

There are natural homomorphisms w : 𝔾_{m} → R_{L/K} and Nm_{L/K} : R_{L/K} → 𝔾_{m}. On the group of K-rational points they induce the inclusion K^{×} ↪ L^{×} and the L/K-norm Nm_{L/K} : L^{×} → K^{×} respectively. Let

R′_{L/K} = ker(Nm_{L/K} : R_{L/K} → 𝔾_{m}), R′′_{L/K} = R_{L/K}/w(𝔾_{m}) .

The character group of X^{∗}(R_{L/K}) is a free abelian group with basis elements {χ_{σ}}_{Hom(L,K^{sep})}; the Galois group Gal(K^{sep}/K) operates transitively on the χ_{σ}’s. The restriction of the character χ_{σ} : R_{L/K} → 𝔾_{m} to L^{×} = R_{L/K}(K) is equal to σ. The character group X^{∗}(R′′_{L/K}) of R′′_{L/K} is the subgroup of X^{∗}(R_{L/K}) generated by elements of the form χ_{σ} −χ_{σ_{0}}, where σ_{0} is a fixed element of Hom(L, K^{sep}). The character group of R′_{L/K} is the quotient of X^{∗}(R_{L/K}) by the subgroup ℤ · (∑_{σ∈Hom(L,K^{sep})} χ_{σ}).

(2.5.1) Let L be a finite Galois extension of K such that the κ_{L}/κ is separable. Let Γ = Gal(L/K). The lower numbering filtration of Γ defined in [S2, V §1] is a decreasing filtration

Γ = Γ_{−1} ⊇ Γ_{0} ⊇ Γ_{1} ⊇ ⋯

indexed by integers ≥ −1 such that Γ_{n} = {1} if n is sufficiently large. For each integer m ≥ 0 and each σ ∈ Γ, σ ∈ Γ_{m} if and only if σ operates trivially on 𝒪_{L}/𝔭_{L}^{m+1}. The function i_{Γ} = i_{L/K} : Γ − {1} → ℕ, defined in [S2, V §1], is characterized by the following property: i_{Γ}(σ) = m if and only if σ ∈ Γ_{m+1} and σ ∉ Γ_{m+2}.

§3. The basic estimates

(3.1) Theorem Let 1 → T′ → T → T′′ → 1 be a short exact sequence of tori over K.

(i) For all 1 ≤ k ≤ dim(T′) we have

∑_{1≤j≤k} c_{j}(T, K) ≤ ∑_{1≤j≤k} c_{j}(T′, K) .

(ii) Assume that the residue field κ of 𝒪 is perfect. Then

∑_{i+dim(T′)≤j≤dim(T)} c_{j}(T, K) ≥ ∑_{i≤j≤dim(T′′)} c_{j}(T′′, K)

for all 1 ≤ i ≤ dim(T′′).

Proof. Consider the commutative diagram

\xymatrix{ & Lie(\underline{T′}^{NR}) ⊗_{𝒪} 𝒪_{L} \ar[r] \ar[d]_{_{can_{T′,L/K}}_{∗}} & Lie(\underline{T}^{NR}) ⊗_{𝒪} 𝒪_{L} \ar[r] \ar[d]_{_{can_{T,L/K}}_{∗}} & Lie(\underline{T′}^{NR}) ⊗_{𝒪} 𝒪_{L} \ar[d]_{_{can_{T′′,L/K}}_{∗}} \\ 0 \ar[r] & Lie(\underline{T′_{L}}^{NR}) \ar[r] & Lie(\underline{T_{L}}^{NR}) \ar[r] & Lie(\underline{T′′_{L}}^{NR}) \ar[r] & 0}

where the second row is exact and the first row is a complex which is not necessarily exact. To simplify the notation in the proof, we abbreviate the above diagram to

\xymatrix{ & M′ ⊗_{𝒪} 𝒪_{L} \ar[r] \ar[d] & M ⊗_{𝒪} 𝒪_{L} \ar[r] \ar[d] & M′′ ⊗_{𝒪} 𝒪_{L} \ar[d] \\ 0 \ar[r] & M′_{L} \ar[r] & M_{L} \ar[r] & M′′_{L} \ar[r] & 0 .}

Insert an exact middle row to obtain an enlarged commutative diagram

\xymatrix{ & M′ ⊗_{𝒪} 𝒪_{L} \ar[r] \ar[d] & M ⊗_{𝒪} 𝒪_{L} \ar[r] \ar[d]^{=} & M′′ ⊗_{𝒪} 𝒪_{L} \\ 0 \ar[r] & M̃′ \ar[r] \ar[d] & M ⊗_{𝒪} 𝒪_{L} \ar[d] \ar[r] & M̃′′ \ar[r] \ar[d] \ar[u] & 0 \\ 0 \ar[r] & M′_{L} \ar[r] & M_{L} \ar[r] & M′′_{L} \ar[r] & 0}

where M̃′ = (M′⊗_{𝒪} 𝒪_{L})∩(M ⊗_{𝒪} 𝒪_{L}), and M̃′′ is equal to the image of M ⊗_{𝒪} 𝒪_{L} in M′′⊗_{𝒪} 𝒪_{L}. Notice that all vertical arrows are injective homomorphisms between finite free 𝒪_{L}-modules of the same rank. The second and the third row are exact, while the first row is a complex of 𝒪_{L}-modules. Denote by r (resp. s) the rank of the free 𝒪-module M′ (resp. M′′), so that M has rank r + s.

Let k be a natural number such that 0 ≤ k ≤ r = dim(T′). Then the natural map Λ_{𝒪_{L}}^{k} (M̃′) → Λ_{𝒪_{L}}^{k} (M ⊗_{𝒪} 𝒪_{L}) is a split injection. Therefore α_{k}(M̃′, M′_{L}) ≥ α_{k}(M ⊗_{𝒪} 𝒪_{L}, M_{L}). Combined with the trivial estimate α_{k}(M′ ⊗_{𝒪} 𝒪_{L}, M′_{L}) ≥ α_{k}(M̃′, M′_{L}), we get α_{k}(M′ ⊗_{𝒪} 𝒪_{L}, M′_{L}) ≥ α_{k}(M ⊗_{𝒪} 𝒪_{L}, M_{L}). Dividing the above inequality by e(L/K) finishes the proof of 3.1 (i).

The proof of 3.1 (ii) is somewhat long and will be divided into several Lemmas for clarity.

(3.1.1) Lemma Let r = rank(M′) = dim(T′), s = rank(M′′) = dim(T′′) as above. Then

α_{r}(M′ ⊗_{𝒪} 𝒪_{L}, M̃′) = α_{s}(M̃′′, M′′ ⊗_{𝒪} 𝒪_{L})

Proof. According to [CYdS, 11.3, 12.1] or [dS] we have c(T′, K) + c(T′′, K) = c(T, K), or equivalently,

α_{r}(M′ ⊗_{𝒪} 𝒪_{L}, M′_{L}) + α_{s}(M′′ ⊗_{𝒪} 𝒪_{L}, M′′_{L}) = α_{r+s}(M ⊗_{𝒪} 𝒪_{L}, M_{L})

On the other hand the exactness of the second and the third row in the digram (*) yields

α_{r+s}(M ⊗_{𝒪} 𝒪_{L}, M_{L}) = α_{r}(M̃′, M′_{L}) + α_{s}(M̃′′, M′′_{L}) .

The Lemma follows.

(3.1.2) Lemma For i = 1, …, s = rank(M′′), we have

α_{i}(M̃′′, M′′_{L}) ≤ α_{i}(M′′ ⊗_{𝒪} 𝒪_{L}, M′′_{L}) + α_{s}(M̃′′, M′′ ⊗_{𝒪} 𝒪_{L})

Proof. This is a special case of the following general statement in linear algebra: Suppose that N_{0} ⊂ N_{1} ⊂ N_{2} are three 𝒪_{L}-lattices in an s-dimensional L-vector space V, then

α_{i}(N_{0}, N_{2}) ≤ α_{i}(N_{1}, N_{2}) + α_{s}(N_{0}, N_{1}) ∀ i = 1, …, s .

To prove this claim, choose an 𝒪_{L}-basis w_{1}, …, w_{s} of N_{1} such that π_{L}^{e_{1}}w_{1}, …, π_{L}^{e_{s}}w_{s} is an 𝒪_{L}-basis of N_{0}. for suitable natural numbers e_{1}, …, e_{s}. Let v_{1}, …, v_{s} be an 𝒪_{L}-basis of N_{2}. Write w_{k} = ∑_{l=1}^{s} a_{kl} v_{l} for k = 1, …, s, so we have π_{L}^{e_{k}}w_{k} = ∑_{l=1}^{s} π_{L}^{e_{k}}a_{kl} v_{l} for each k.

By definition, the determinant of every i × i-minor of the s × s-matrix B = (π_{L}^{e_{k}}a_{kl}) ∈ M_{s}(𝒪_{L}) is divisible by π_{L}^{α_{i}(N_{0},N_{2})}. Hence the determinant of every i × i-minor of the s × smatrix A = (a_{kl}) ∈ M_{s}(𝒪_{L}) is divisible by π_{L}^{α_{i}(N_{0},N_{2})} − ∑_{k=1}^{s} e_{k}. This means that

α_{i}(N_{0}, N_{2}) − α_{s}(N_{0}, N_{1}) = α_{i}(N_{0}, N_{2}) − ∑_{k=1}^{s} e_{k} ≤ α_{i}(N_{1}, N_{2}) .

The claim is proved. Notice that the argument also shows that we can replace α_{s}(N_{0}, N_{1}) = ∑_{k=1}^{s} e_{k} in the claim by

max_{I⊆{1,... ,s}, Card(I)=i} (∑_{k∈I} e_{k})

in the claim.

(3.1.3) Lemma Notation as above, in particular r = rank(M′). Then

α_{r+i}(M ⊗_{𝒪} 𝒪_{L}, M_{L}) ≤ α_{r}(M̃′, M′_{L}) + α_{i}(M̃′′, M′′_{L})

for i = 1, …, s = rank(M′′).

Proof. This is a consequence of the fact that the natural map

Λ_{𝒪_{L}}^{r} (M̃′) ⊗_{𝒪_{L}} Λ_{𝒪_{L}}^{i} (M̃′′) → Λ_{𝒪_{L}}^{r+i}(M ⊗_{𝒪} 𝒪_{L})

is a split injection.

Proof of 3.1 (ii), continued. Let i be a natural number such that 0 ≤ i ≤ s = dim(T′′). Then

α_{r+i}(M ⊗_{𝒪} 𝒪_{L}, M_{L}) ≤ α_{r}(M̃′, M′_{L}) + α_{i}(M̃′′, M′′_{L}) by Lemma 3.1.3
= α_{r}(M′ ⊗_{𝒪} 𝒪_{L}, M′_{L}) − α_{r}(M′ ⊗_{𝒪} 𝒪_{L}, M̃′) + α_{i}(M̃′′, M′′_{L})
= α_{r}(M′ ⊗_{𝒪} 𝒪_{L}, M′_{L}) − α_{s}(M̃′′, M′′ ⊗_{𝒪} 𝒪_{L}) + α_{i}(M̃′′, M′′_{L}) by Lemma 3.1.1
≤ α_{r}(M′ ⊗_{𝒪} 𝒪_{L}, M′_{L}) + α_{i}(M′′ ⊗_{𝒪} 𝒪_{L}, M′′_{L}) by Lemma 3.1.2

Dividing the above inequality by e(L/K) gives

∑_{1≤j≤dim(T′)+i} c_{j}(T, K) ≤ c(T′K) + ∑_{1≤j≤i} c_{j}(T′′, K)

Since c(T, K) = c(T′, K) + c(T′′, K), the above inequality is equivalent to

∑_{dim(T′)+i+1≤j≤dim(T)} c_{j}(T, K) ≥ ∑_{i+1≤j≤dim(T′′)} c_{j}(T′′, K) .

We have finished the proof of Thm. 3.1 (ii).

(3.1.4) Remark Theorem 3.1 was asserted without proof in [Ch1] 8.5 (d). Unfortunately 3.1 (ii) appeared incorrectly there.

(3.2) Proposition Let 1 → T′ → T → T′′ → 1 be an exact sequence of tori over K. Assume that T′ is a split torus over K.

(i) The induced complex

0 → Lie(\underline{T′}^{NR}) → Lie(\underline{T}^{NR}) → Lie(\underline{T′′}^{NR}) → 0

is a short exact sequence of 𝒪-modules.

(ii) For each i = 1, …, dim(T′′), we have c_{dim(T′)+i}(T, K) = c_{i}(T′′).

Proof. The statement (i) is proved in [Ch1] 4.5 and 4.8. Statement (ii) follows from (i).

(3.3) Proposition Let T be a torus over K which is anisotropic over the maximal unramified extension K^{sh} of K.

(i) The neutral component of the closed fiber of \underline{T}^{NR} is a unipotent commutative algebraic group over κ.

(ii) We have c_{i}(T, K) > 0 for i = 1, …, dim(T).

Proof. We may and do assume that 𝒪 is strictly henselian. Since T is anisotropic, T (K) is bounded: Every element x ∈ T (K) corresponds to a Γ-equivariant homomorphism h_{x} : X^{∗}(T) → L^{×}, and ord_{L} ∘h_{x} : X^{∗}(T) → ℤ must be trivial because the Γ-coinvariant of X^{∗}(T) is finite. Therefore \underline{T}^{NR} = \underline{T}^{lft NR}.

Let \underline{T}^{NR}_{κ}^{∘} be the neutral component of \underline{T}^{NR} × Spec 𝒪 Spec κ. It is known that every connected commutative algebraic group over κ is canonically isomorphic to a product of a commutative unipotent group over κ and a torus over κ. Write \underline{T}^{NR}_{κ}^{∘} = U × W, where U is a commutative unipotent group over κ and W is a torus over κ. Let ℓ be a prime number different from p = Char(κ). For every natural number n, [ℓ^{n}]_{U} : U → U is an isomorphism, [ℓ^{n}]\underline{T}_{NR}_{κ}^{∘} is étale, and [ℓ^{n}]_{W} : W → W is an étale isogeny. The kernel–cokernel long exact sequence gives an isomorphism \underline{T}^{NR}_{κ}^{∘}[ℓ^{n}] ⟶^{∼} W [ℓ^{n}] between the two finite étale group schemes over κ. Since [ℓ^{n}]\underline{T}^{NR} : \underline{T}^{NR} → \underline{T}^{NR} is étale and 𝒪 is strictly henselian, every element of \underline{T}^{NR}_{κ}^{∘}[ℓ^{n}] can be uniquely lifted to an element of \underline{T}^{NR}[ℓ^{n}](𝒪). So if W is nontrivial, the ℓ-adic Tate module V(T) attached to T has a non-trivial Γ-invariant element. This is impossible because V(T) and X_{∗}(T) ⊗_{ℤ} ℚ_{ℓ}(1) are isomorphic as Γ-modules, and (X_{∗}(T) ⊗_{ℤ} ℚ)^{Γ} is assumed to be trivial. We have proved (i).

Consider the map between Lie algebras (can_{T,L/K})_{∗} induced by the homomorphism can_{T,L/K} : \underline{T}^{NR} ×_{Spec 𝒪} Spec 𝒪_{L} → \underline{T_{L}}^{NR}. Its reduction modulo 𝔭_{L} is canonically identified with the tangent map of the restriction of can_{T,L/K} to the closed fibers. The neutral component of the closed fiber of \underline{T}^{NR} is unipotent by (i), and the closed fiber of \underline{T_{L}}^{NR} is a split torus by definition. So can_{T,L/K} is trivial on the neutral component of \underline{T}^{NR} ×_{Spec 𝒪} Spec 𝒪_{L}, and its tangent map is trivial. This proves (ii).

(3.3.1) Remark (i) Here is an alternative proof of (ii) Suppose that L is a finite Galois extension of K such that T is split over K. We may and do assume that L is totally ramified over K, in the sense that κ_{L} is a purely inseparable extension of κ. By Prop. 3.4 (ii), it suffices to verify the statement of 3.3 for T = R′_{L/K}. According to the description of Lie(\underline{T}^{NR} in [dS, A1.7], we must show that if u(x) = 1 + ∑_{n≥1} a_{n} x^{n} is a formal power series 𝒪_{L}[[x]] such that Nm_{L/K}(u(x)) = 1, then a_{1} ∈ 𝔭_{L}. Let b_{n} be the image of a_{n} in κ_{L}, and let v(x) = 1+∑_{n≥1} b_{n} x^{n} be the image of u(x) in κ_{L}[[x]]. We have 1 = Nm_{κ_{L}/κ}(v(x)) = v(x)^{[L:K]}. Hence v(x) = 1. In particular a_{1} ∈ 𝔭_{L}.

(ii) If κ_{L} is separable over κ, the same line of argument provides an explicit lower bound on c_{1}(T, K); see Cor. 4.5.

(3.4) Proposition Let T be a torus over K, and let L be a finite Galois extension of K such that T is split over K. Assume that T is anisotropic over K.

(i) There exists a short exact sequence of tori of the form

1 → T′ → (R′′_{L/K})^{m} → T → 1

(ii) There exists a short exact sequence of tori of the form

1 → (R′_{L/K})^{n} → T → T′′ → 1

Proof. We first prove (i). The cocharacter group X_{∗}(T) has a natural structure as a module over the finite group Γ = Gal(L/K). The assumption that T is anisotropic over K means that X_{∗}(T)^{Γ}, the submodule of all Γ-invariants in X_{∗}(T), is trivial. Choose a Γ-equivariant surjection h : ℤ[Γ]^{⊕m} ↠ X_{∗}(T). The submodule (ℤ · ∑_{σ∈Γ} σ)^{⊕m} ⊂ (ℤ[Γ]^{⊕m})^{Γ} maps to zero under h, because X_{∗}(T)^{Γ} = (0). The resulting Γ-equivariant surjection

(ℤ[Γ]/ℤ · ∑_{σ∈Γ} σ)^{⊕m} ↠ X_{∗}(T)

gives a homomorphism of tori required in (i). The same argument, using the character group X^{∗}(T) instead of the cocharacter group X_{∗}(T), proves (ii).

§4. Induced tori and norm-one tori

(4.1) Proposition Let L be a totally ramified finite Galois extension of K such that κ_{L} = κ. Let Γ = Gal(L/K). Then for each k = 1, …, e(L/K), we have

c_{1}(R_{L/K}, K) + ⋯ + c_{k}(R_{L/K}, K) = min_{I⊆Γ, Card(I)=k} (\frac{1}{2 e(L/K)} ∑_{{σ,τ∈Γ \atop σ≠τ}} i_{Γ}(σ τ^{−1})) .

In particular, c_{1}(R_{L/K}, K) = 0, and ∑_{i=1}^{e(L/K)} c_{i}(R_{L/K}, K) = \frac{1}{2} ord_{K}(disc(L/K)).

Proof. Let Γ = Gal(L/K) = {1 = σ_{1}, …, σ_{n}}, n = [L : K] = e(L/K). Let π_{L} be a generator of 𝔭_{L}. As an 𝒪-algebra, 𝒪_{L} is generated by π_{L}. Hence i_{Γ}(σ_{i} σ_{j}^{−1}) = ord_{L}(σ_{i}(π_{L}) − σ_{j}(π_{L})) for any 1 ≤ i ≠ j ≤ n. Let A be the n × n matrix whose (i, j)-th entry is σ_{j}(π_{L}^{i−1}) for i, j = 1, …, n. For a natural number k, 1 ≤ k ≤ n, n · (c_{1}(R_{L/K}, K) + ⋯ + c_{k}(R_{L/K}, K)) is equal to the minimum among the orders (measured by ord_{L}) of the determinant of all k × k-minors of A; we have to show that it is equal to

min_{I⊆\{1,... ,n\}, Card(I)=k} (∑_{{i<j \atop i,j∈I}} ord_{L}(σ_{i}(π_{L}) − σ_{j}(π_{L}))) .

That each sum ∑_{{i<j \atop i,j∈I}} ord_{L}(π_{L}^{i} − π_{L}^{j}) in the above displayed formula is equal to the order of the determinant of a k × k-minor of A follows from the Vandermonde determinant. The following Lemma 4.1.1 finishes the proof.

(4.1.1) Lemma Let m be a positive integer, and let a_{1}, …, a_{m} ≥ 0 be mutually distinct natural numbers. Consider the m × m-matrix B with entries in the polynomial ring ℤ[x_{1}, …, x_{m}] whose (i, j)-th entry is equal to x_{j}^{a_{i}} for i, j = 1, …, m. Then det(B) is divisible by ∏_{1<i,j<m} (x_{i} − x_{j}) in ℤ[x_{1}, …, x_{m}]

Proof of 4.1.1. Clearly det(B) is divisible by (x_{i}−x_{j}) if i ≠ j. The Lemma follows because ℤ[x_{1}, …, x_{m}] is a unique factorization domain. This finishes the proof of Lemma 4.1.1 and Prop. 4.1.

(4.2) Corollary Notation as above, and we assume that L ≠ K.

(i) Let t be the natural number such that Γ = Γ_{t}, Γ_{t} ≠ Γ_{t+1}. Then

c_{i}(R′′_{L/K}, K) = c_{i+1}(R_{L/K}, K) = \frac{i(t + 1)}{e(L/K)}

for i = 1, …, [Γ : Γ_{t+1}] − 1.

(ii) We have c_{e(L/K)−1}(R′′_{L/K}, K) = c_{e(L/K)}(R_{L/K}, K) = \frac{ord_{K}(disc(L/K))}{e(L/K)}. More generally,

c_{e(L/K)−i}(R′′_{L/K}, K) = c_{e(L/K)−i+1}(R_{L/K}, K) = \frac{ord_{K}(disc(L/K))}{e(L/K)} − \frac{(i − 1)(t + 1)}{e(L/K)}

for i = 1, …, [Γ : Γ_{t+1}].

Proof. (i) Let a = [Γ : Γ_{t+1}]. We may and do assume that {1 = σ_{1}, …, σ_{a}} is a set of representatives of Γ/Γ_{t+1}. Then for k = 1, …, a} the minimum in the formula in 4.1 for c_{1}(R_{L/K}, K) + … + c_{k}(R_{L/K}, K) is reached with I being any subset of {σ_{1}, …, σ_{a}} with k elements; it is equal to \frac{k(k−1)(t+1)}{2e(L/K)}. Hence c_{k}(R_{L/K}, K) = \frac{(k−1)(t+1)}{e(L/K)} for k = 1, …, a.

(ii) For i = 1, …, [Γ : Γ_{t+1}], the minimum in the formula in 4.1 for c_{1}(R_{L/K}, K) + … + c_{e(L/K)−i}(R_{L/K}, K) is reached when Γ − I is a subset of {σ_{1}, …, σ_{a}} with i elements, where {σ_{1}, …, σ_{a}} is a set of representatives of Γ/Γ_{t+1} as in (i). We fix a subset J_{0} = {τ_{1}, …, τ_{i}} of Γ_{s} with i elements, let I_{0} = Γ − J_{0}, and use I_{0} to compute the minimum. One finds that

c_{e(L/K)−i+1}(R_{L/K}, K) + … + c_{e(L/K)}(R_{L/K}, K)
= \frac{i}{e(L/K)} ∑_{1≠σ∈Γ} i_{Γ}(σ) − \frac{1}{e(L/K)} ∑_{1≤k<l≤i} i_{Γ}(σ_{k} · σ_{l}^{−1})
= \frac{i}{e(L/K)} ord_{K}(disc(L/K)) − \frac{i(i−1)(t+1)}{2 e(L/K)} .

The statement (ii) follows.

(4.3) Corollary Assume that the residue field κ of 𝒪 is perfect. Let T be a torus over K which splits over a finite Galois extension L of K. Then

c_{dim(T)}(T, K) ≤ \frac{ord_{K}(disc(L/K))}{e(L/K)} .

Proof. This follows from Prop. 3.2, Prop. 3.4 (i), Thm. 3.1 (ii) and Cor. 4.2 (ii).

(4.4) Theorem Let L be a totally ramified finite Galois extension of K such that κ_{L} = κ. Let Γ = Gal(L/K), and let Γ = Γ_{0} ⊇ Γ_{1} ⊇ ⋯ be the lower-numbering filtration of Γ. Let t be the natural number such that Γ = Γ_{t} ≠ Γ_{t+1}. Then c_{1}(R′_{L/K}, K) = \frac{t+1}{e(L/K)}.

Proof. By Prop. 3.4 (ii), Thm. 3.1 (i) and Cor. 4.2 (i), we have

c_{1}(R′_{L/K}, K) ≤ c_{1}(R′′_{L/K}, K) = \frac{t + 1}{e(L/K)} .

We will prove the inequality c_{1}(R′_{L/K}, K) ≥ \frac{t+1}{e(L/K)} in the reversed direction by the method in [S2] V §3–§7; we have already used a part of it in the proof of 3.3.1. According to the description of Lie(\underline{T}^{NR}) in [dS, A1.7], the Lie algebra of \underline{R′_{L/K}}^{NR} can be identified with all elements a_{1} ∈ 𝒪_{L} such that there exists a formal power series of the form

u(x) = 1 + a_{1} x + ∑_{n≥2} a_{n} x^{n} ∈ 𝒪_{L}[[x]]^{×}

such that Nm_{L/K}(u(x)) = 1. We claim that a_{n} ∈ 𝔭_{L}^{t+1} for all n ≥ 1 for any u(x) as above. Of course this claim implies that the inequality c_{1}(R′_{L/K}, K) ≥ \frac{t+1}{e(L/K)} we need for the proof of 4.4. This will be accomplished in a series of lemmas below; the claim itself is the statement of Lemma 4.4.5.

We introduce some notation. Let U_{L,0} be the subgroup of 𝒪_{L}[[x]]^{×} consisting of all elements of the form u(x) = 1 + ∑_{m≥1} a_{m} x^{m}, a_{m} ∈ 𝒪_{L} for all m ≥ 1. For each integer b ≥ 1, denote by U_{L,b} the subgroup of U_{L,0} consisting of all elements of the form u(x) = 1 + ∑_{m≥1} a_{m} x^{m} such that a_{m} ∈ 𝔭_{L}^{b} for all m ≥ 1. The quotient U_{L,0}/U_{L,1} is canonically isomorphic to the subgroup 1 + x · κ[[x]] of κ[[x]]^{×}. Choose a generator π_{L} of 𝔭_{L}, we obtain an isomorphism β_{b,π_{L}} : U_{L,b}/U_{L,b+1} ⟶^{∼} x · κ[[x]] for each b ≥ 1. This map β_{b,π_{L}} sends an element u(x) = 1 + π_{L}^{b} ∑_{m≥1} a_{m} x^{m} of U_{L,b} to the element ∑_{m≥1} \overline{a_{m}}x^{m} of x · κ[[x]], where \overline{a_{m}} is the image of a_{m} ∈ 𝒪_{L} in κ.

(4.4.1) Lemma Suppose that Γ = Gal(L/K) is cyclic of order ℓ, where ℓ is a prime number, and κ_{L} = κ. Let σ be a generator of Γ. Suppose that i_{Γ}(σ) = t+1, so that Γ = Γ_{t}, Γ_{t+1} = {1}. Then for each n ≥ 0, we have

Tr_{L/K}(π_{L}^{n} · x𝒪_{L}[[x]]) = π_{K}^{r} · x𝒪_{L}[[x]] where r = ⌊\frac{n + (t + 1)(ℓ − 1)}{ℓ}⌋ .

In particular,

\begin{aligned}Tr_{L/K}(𝔭_{L}^{n} x · 𝒪_{L}[[x]]) & ⊆ & 𝔭^{n+1} x · 𝒪[[x]] for n = 1, …, t − 1 \\ Tr_{L/K}(𝔭_{L}^{t} x · 𝒪[[x]]) & = & 𝔭^{t} x · 𝒪[[x]] \end{aligned}

Proof. This follows from [S2, V §3, Lemma 3].

(4.4.2) Lemma Notation as in 4.4.1. Suppose that ξ(x) ∈ 𝔭_{L}^{n} x · 𝒪_{L}[[x]], n ≥ 1. Then

Nm_{L/K}(1 + ξ(x)) ≡ 1 + Tr_{L/K}(ξ(x)) + Nm_{L/K}(ξ(x)) (mod Tr_{L/K}(𝔭_{L}^{2n})) .

Proof. The argument in [S2, V §3 Lemma 5] works here as well.

(4.4.3) Lemma Notation as in 4.4.1.

(i) We have Nm_{L/K}(U_{L,n}) ⊆ U_{K,n} for each 0 ≤ n ≤ t + 1.

(ii) Let u(x) be an element of U_{L,0}. Let \overline{u(x)} be the image of u(x) in 1+x·κ[[x]] ≅ U_{L,0}/U_{L,1}. Then the image of Nm_{L/K}(u(x)) in 1 + x · κ[[x]] ≅ U_{K,0}/U_{K,1} is \overline{u(x)}^{ℓ}.

(iii) Let u(x) be an element of U_{L,n}, 1 ≤ n ≤ t − 1. Let \overline{ξ(x)} = β_{n,π_{L}}(u(x)) ∈ x · κ[[x]]. Then the image of Nm_{L/K}(u(x)) in x · κ[[x]] under β_{n,π_{K}} is α_{n} \overline{ξ(x)}^{p}, where α_{n} denotes the image of \frac{Nm_{L/K}(π_{L}^{n})}{π_{K}^{n}} ∈ 𝒪_{L}^{×} in κ^{×}.

(iv) The map U_{L,t}/U_{L,t+1} → U_{K,t}/U_{K,t+1} induced by Nm_{L/K} can be described as follows. Identify U_{L,t}/U_{L,t+1} with x · κ[[x]] via β_{t,π_{L}}, and identify U_{K,t}/U_{K,t+1} with x · κ[[x]] via β_{t,π_{K}}. Then there exists α, γ ∈ κ^{×} such that Nm_{L/K} sends \overline{ξ(x)} ∈ x·κ[[x]] ≅ U_{L,t}/U_{L,t+1} to α \overline{ξ(x)}^{p} + γ \overline{ξ(x)}.

Proof. Use 4.4.2, 4.4.1 and the argument of [S2] V §3, Prop. 4 and Prop. 5.

(4.4.4) Lemma Let L be a finite Galois extension of K with group Γ, and assume that κ_{L} = κ. Suppose that Γ = Γ_{t} ⊋ Γ_{t+1}.

(i) The L/K-norm satisfies Nm_{L/K}(U_{L,n}) ⊂ U_{K,n} for 0 ≤ n ≤ t + 1, and induces a homomorphism Nm_{L/K,n} : U_{L,n}/U_{L,n+1} → U_{K,n}/U_{K,n+1} for n = 0, 1, …, t.

(ii) Suppose that 1 ≤ n ≤ t. Identify U_{L,n}/U_{L,n+1} with x · κ[[x]] via β_{n,π_{L}}, and identify U_{K,n}/U_{K,n+1} with x · κ[[x]] via β_{n,π_{K}}. Then

Nm_{L/K,n} : x · κ[[x]] ⟵_{β_{n,π_{L}}}^{∼} U_{L,n}/U_{L,n+1} → U_{K,n}/U_{K,n+1} ⟶_{β_{n,π}}^{∼} x · κ[[x]]

can be described as follows. If 1 ≤ n ≤ t − 1, then there exists an element δ_{n} ∈ κ^{×} such that

Nm_{L/K,n} : \overline{ξ(x)} ↦ δ_{n} · \overline{ξ(x)}^{[L:K]} ∀ \overline{ξ(x)} ∈ x · κ[[x]] .

(iii) Notation as in (ii). For n = t, there exists an additive polynomial P (Y) ∈ κ[Y] of degree [L : K] and separable degree [Γ : Γ_{t+1}] such that

Nm_{L/K,t} : \overline{ξ(x)} ↦ P (ξ(x)) ∀ \overline{ξ(x)} ∈ x · κ[[x]] .

Proof. The proof is similar to that of [S2] V §6 Prop. 9. One can build up the Galois extension L/K, starting from K, by a sequence of successive cyclic extensions. Apply Lemma 4.4.3 and use induction.

(4.4.5) Lemma Notation as in 4.4.4 Suppose that u(x) is an element of U_{L,0} such that Nm_{L/K}(u(x)) = 1. Then u(x) ∈ U_{L,t+1}.

Proof of 4.4.5 and Thm. 4.4. As explained at the beginning of the proof of 4.4, Lemma 4.4.5 implies Thm. 4.4. We have seen in 3.3.1 that u(x) ∈ U_{L,1}. Apply Lemma 4.4.4 (ii) repeatedly, we deduce that u(x) ∈ U_{L,t}. It remains to show that u(x) ∈ U_{L,t+1}. Let \overline{ξ(x)} = β_{t,π_{L}}. Apply Lemma 4.4.4 (iii) and use the notation there, we have P (\overline{ξ(x)}) = 0 in κ[[x]], where \overline{ξ(x)} ∈ x · κ[[x]] is the image of u(x) under β_{t,π_{L}} : U_{L,t}/U_{L,t+1} ⟶^{∼} x · κ[[x]]. We want to deduce, from the equation P (\overline{ξ(x)}) = 0, the desired conclusion that \overline{ξ(x)} = 0 in κ[[x]].

There exists an additive polynomial P_{s}(T) ∈ κ^{alg}[T] of the form

P_{s}(T) = b_{k} T^{p^{k}} + ⋯ + b_{1} T^{p} + a_{0} T with b_{0} ∈ (κ^{alg})^{×}

such that P (T) = P_{s}(T^{Card(Γ_{t+1})}). Notice that Card(Γ_{t+1}) is a power of p. We know that P_{s}(\overline{ξ(x)}^{Card(Γ_{t+1})}) = P (\overline{ξ(x)}) = 0. Therefore \overline{ξ(x)}^{Card(Γ_{t+1})} = 0 by Hensel’s Lemma. Hence \overline{ξ(x)} = 0.

(4.5) Corollary Let L be a finite Galois extension of K such that κ_{L}/κ_{K} is separable. Let T be a torus over K which is split over L. Assume that T is anisotropic over the maximal unramified extension of K. Then c_{1}(T, K) ≥ \frac{t+1}{e(L/K)}, where t is the natural number such that Γ = Γ_{t} ≠ Γ_{t+1}.

Proof. This follows from Prop. 3.4 (ii), Thm. 3.1 and Prop. 4.4.

§5. Examples

(5.1) Example Let L/K be a totally ramified extension such that Γ = Gal(L/K) is cyclic of order p. Assume that κ_{L} = κ and Char(κ) = p.

(i) For any two anisotropic torus T_{1}, T_{2} over K of with dim(T_{1}) = dim(T_{2}), there exists an isogeny α : T_{1} → T_{2} of degree prime to p. Consequently c_{i}(T_{1}, K) = c_{i}(T_{2}, K) for each i = 1, …, dim(T_{1}).

(ii) Let t be the integer such that Γ = Γ_{t}, Γ_{t+1} = {1}. Let T be an anisotropic torus over K of dimension p − 1. Then c_{i}(T, K) = \frac{i(t+1)}{p} for i = 1, …, p − 1.

Proof. (i) The quotient of the group ring ℤ[Γ] by the ideal generated by the element ∑_{σ∈Γ} σ is isomorphic to ℤ[x]/(x^{p−1} + … + x + 1), the ring of integers in the cyclotomic field ℚ(μ_{p}). Hence the localization R_{(p)} of the ring R := ℤ[Γ]/(∑_{σ∈Γ} σ) at the ideal generated by p is a discrete valuation ring. For i = 1, 2, the character group X_{∗}(T_{i}) is naturally an R-module. Hence X_{∗}(T_{1})_{(p)} is isomorphic to X_{∗}(T_{2})_{(p)} as R_{(p)}-modules. In other words there exists a Γ-equivariant homomorphism h : X_{∗}(T_{1}) → X_{∗}(T_{2}) such that ker(h) and coker(h) are finite groups of order prime to p.

(ii) By (i) we may and do assume that T = R′′_{L/K}, so we can apply 4.2 (i).

(5.2) Example Let m ≥ 2 be a natural number. Let K = ℚ_{2}(μ_{2^{m}}), M = ℚ_{2}(μ_{2^{m+1}}), and L = ℚ_{1}(μ_{2^{m+2}}). Let T′ = R_{M/K}, T = R_{L/K}, and let T′′ be the quotient torus T/T′. Then

\underline{c}(T′, K) = (0, 2^{m−1}), \underline{c}(T′′, K) = (2^{m−1}, 2^{m}), \underline{c}(T, K) = (0, 2^{m−2}, 3 · 2^{m−2}, 2^{m}) .

Proof. The Galois group Γ = Gal(L/K) is cyclic of order 4, canonically isomorphic to the subgroup of ℤ/2^{m+2}ℤ)^{×} generated by the image of 1 + 2^{m}; denote this element by σ. Let ζ be a primitive 2^{m+2}-th root of unity. Then

i_{Γ}(σ) = ord_{L}(σ(ζ) − ζ) = ord_{L}(ζ · (\sqrt{−1} − 1)) = 2^{m}

By 4.2 (i) we get c_{2}(T, K) = 2^{m−2}. From 4.2 (ii) we get c_{4}(T, K) = 2^{m}, and c_{3}(T, K) = 2^{m} − 2^{m−2} = 3 · 2^{m−2}.

Let Δ = Gal(M/K), generated by the image σ̄ of σ in Δ. Then

ı_{Δ}(σ̄) = ord_{M}(σ̄(ζ^{2}) − ζ^{2}) = ord_{M}(ζ^{2} · (−2)) = 2^{m}

and c_{1}(T′, K) = 2^{m−1} by 4.2 (i).

We still have to compute \underline{c}(T′′, K). By [Ch1] 4.5 and 4.8, we have a natural short exact sequence

0 → Lie(\underline{T′}^{NR}) → Lie(\underline{T}^{NR}) → Lie(\underline{T′′}^{NR}) → 0

which is compatible with the map on Lie algebras induced by the base change map can_{L/K}. We identify Lie(\underline{T′}^{NR}) with 𝒪_{M}, and Lie(\underline{T}^{NR}) with 𝒪_{L}. So Lie(\underline{T′′}^{NR}) is identified with 𝒪_{L}/𝒪_{M}. We identify X^{∗}(T) with ℤ[Γ], and the character group X^{∗}(T′′) of T′′) is identifies with the subgroup ℤ · (σ^{2} − 1) + ℤ · (σ^{3} − σ) of ℤ[Γ]. We use the above choice of basis of X^{∗}(T′′) to identify Lie(\underline{T′′_{L}}^{NR}) with 𝒪_{L} ⊕ 𝒪_{L}. Hence the map

can_{T′′,L/K}_{∗} : Lie(\underline{T′′}^{NR}) ⊗_{𝒪} 𝒪_{L} → Lie(\underline{T′′_{L}}^{NR})

is identified with the map from (𝒪_{L}/𝒪_{M}) ⊗_{𝒪} 𝒪_{L} → 𝒪_{L} ⊕ 𝒪_{L} which sends each element ȳ ∈ 𝒪_{L}/𝒪_{M} represented by y ∈ 𝒪_{L} to the element (σ^{2}(y) − y, σ^{3}(y) − σ(y)) ∈ 𝒪_{L} ⊕ 𝒪_{L}. Using the 𝒪-basis {ζ̄, ζ̄^{3}} of 𝒪_{L}/𝒪_{M}, one finds that the elementary divisors of can_{T′′,L/K}_{∗} are (2, 4). In other words c_{1}(T′′, K) = ord_{K}(2) = 2^{m−1}, c_{2}(T′′, K) = ord_{K}(4) = 2^{m}.

(5.3) Example This example shows that if the residue field κ of 𝒪 is not perfect, then the base change conductor c(T, K) for tori may change under K-isogenies.

The polynomial ring ℤ[u, v] contains subrings ℤ[x, v], ℤ[u, y], and ℤ[x, uv], where x = u^{2}, y = v^{2}. Let 𝒪_{K} be the (2)-adic completion of ℤ[x, y], 𝒪_{M_{1}} be the (2)-adic completion of ℤ[u, y], 𝒪_{M_{2}} be the (2)-adic completion of ℤ[x, v], 𝒪_{M_{3}} be the (2)-adic completion of ℤ[x, uv], and let 𝒪_{L} be the (2)-adic completion of ℤ[u, v]. These are complete discrete valuation rings, and 2 is a uniformizing element in each of them. The residue field of 𝒪_{L} (resp. 𝒪_{M_{1}}, 𝒪_{M_{2}}, 𝒪_{M_{3}}, 𝒪_{K}) is 𝔽_{2}(u, v) (resp. 𝔽_{2}(u, y), 𝔽_{2}(x, v), 𝔽_{2}(x, uv), 𝔽_{2}(x, y).) Denote by K, M_{1}, M_{2}, M_{3}, L the fraction field of 𝒪_{K}, 𝒪_{M_{1}}, 𝒪_{M_{2}}, 𝒪_{M_{3}} respectively. Let T = R_{L/K}, and let T_{i} = R_{M_{i}/K}, T′′_{i} = T/T_{i}, i = 1, 2, 3. Then

\underline{c}(T, K) = (0, 1, 1, 2), \underline{c}(T_{i}, K) = (0, 1), \underline{c}(T′′_{i}, K) = (1, 2) .

On the other hand, let T′_{i} = R′_{M_{i}/K} ≅ R′′_{M_{i}/K}, i = 1, 2, 3. Each T′_{i} is a one-dimensional torus over K, and c(T′_{i}, K) = 1 for i = 1, 2, 3. Notice that T′′_{1} is isogenous to T′_{2} ×T′_{3}, c(T′′_{1}, K) = 3, c(T′′_{2} × T′′_{3}, K) = 2.

Sketch of proof. The Galois group Gal(L/K) is isomorphic to (ℤ/2ℤ)^{2}. Let σ (resp. τ) be the element of Gal(L/K) such that σ(u) = −v, σ(v) = v (resp. τ(u) = u, τ(v) = −v), so Gal(L, K) = {1, σ, τ, στ}. The set {1, u, v, uv} is an 𝒪_{K}-basis of 𝒪_{L}. So the c_{i}(T, K)’s are the elementary divisors of the matrix

( \begin{matrix}1 & 1 & 1 & 1 \\ u & −u & u & −u \\ v & v & −v & −v \\ uv & −uv & −uv & uv \end{matrix} ) .

A simple computation shows that \underline{c}(T, K) = (0, 1, 1, 2). A similar but simpler calculation shows that \underline{c}(T_{i}, K) = (0, 1), i = 1, 2, 3. The Lie algebra of \underline{T′′_{i}}^{NR} is naturally isomorphic to the quotient of the Lie algebra of \underline{T}^{NR} by the Lie algebra of \underline{T_{i}}^{NR} according to [Ch1] 4.5 and 4.8, so c_{1}(T′′_{i}, K) + c_{2}(T′′_{i}, K) = 3. On the other hand c_{1}(T′′_{i}, K) > 0 by 3.3. Therefore \underline{c}(T′′_{i}, K) = (1, 2) because there is no other possibility.

(5.4) Example We generalize 5.3 to odd primes. Let p be a prime number. Consider the polynomial ring ℤ[u, v] and its subring ℤ[u^{p}, v^{p}]; let x = u^{p}, y = v^{p}. Let 𝒪_{K} and 𝒪_{L} be the (2)-adic completion of ℤ[x, y] and ℤ[x, y] respectively; denote the fraction fields by K, L. The field L is separable over K of degree p^{2}. Let K_{1} = K(μ_{p}), L_{1} = L(μ_{p}). Then L_{1}/K_{1} is Galois with Gal(L_{1}/K_{1}) ≅ (ℤ/pℤ)^{2}. Let T = R_{L/K}, T_{1} = R_{L_{1}/K_{1}}. Then each c_{i}(T, K) has the form \frac{i}{p−1}, where i is an integer, 0 ≤ i ≤ 2p − 2. For a natural number i with 0 ≤ i ≤ 2p − 2, the multiplicity of \frac{i}{p−1} in \underline{c}(T, K) is equal to min(i + 1, 2p − 1 − i). Similarly each c_{i}(T_{1}, K_{1}) is an integer i with 0 ≤ i ≤ 2p − 2. For a natural number i with 0 ≤ i ≤ 2p − 2, the multiplicity of i in \underline{c}(T_{1}, K_{1}) is equal to min(i + 1, 2p − 1 − i). The proofs of the above assertions are similar to the proof of 5.3, therefore omitted.

It is easy to compute the discriminants: disc(L/K) = (p)^{2p^{2}}, while disc(L_{1}/K_{1}) = (p)^{2p^{2}} = (ζ_{p} −1)^{2(p−1)p^{2}}. Notice that c_{p^{2}}(T_{1}, K_{1}) = 2(p−1), and it coincides with the formula in 4.2 (ii) for c_{dim(T)}(T, K) for an induced torus T, if we interpret “e(L/K)” in 4.2 as [L : K] when κ_{L}/κ_{K} is purely inseparable. This observation still holds if we change ℤ[u, v]/ℤ[u^{p}, v^{p}] to ℤ[u_{1}, …, u_{m}]/ℤ[u_{1}^{p}, …, u_{m}^{p}]: Then disc(L_{1}/K_{1}) = n(p − 1)p^{n}, and c_{p^{n}}(R_{L_{1}/K_{1}}, K_{1}) = n(p − 1). Whether this is an accident, or is a special case of a general phenomenon, seems unclear at present.

References

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[S3] J.-P. Serre. Galois Cohomology. Springer, 1997.

Ching-Li Chai Department of Mathematics University of Pennsylvania Philadelphia, PA 19003 U. S. A.

email: chai@math.upenn.edu
home page: http://www.math.upenn.edu/^{∼}chai