Elementary divisors of the base change conductor for tori. Ching-Li Chai 1 Version 1.0, July 3, 2001

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1 Elementary divisors of the base change conductor for tori 1. Introduction Ching-i Chai 1 Version 1.0, July 3, 2001 et 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) G d (Z) with finite image. et T be a torus over K. The definition of the numerical invariants 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 ie algebra of the Néron model T NR to the ie algebra of the Néron model T NR over O, where is a finite Galois extension of K such that the torus T is split over. One may think of (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 c(t, K) as a numerical invariant c(ρ) attached to a continuous Galois representation ρ : Gal(K sep /K) G 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 G d (Z (p) ), then c(ρ 1 ) = c(ρ 2 ). So one can localize at p and define c(ρ) for any continuous Galois representation ρ : Gal(K sep /K) G 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. et 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 of K. et Γ = Gal(/K) be the Galois group 1 partially supported by grant DMS from the National Science Foundation 1

2 of /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: t + 1 c r+1(t, K) c dim T (T, K) ord K(disc(/K)) ( ) See Cor. 4.3 and Cor In a sense these bounds cannot be improved: Denote by R /M := Res /K (G m ) the Weil restriction of G m from to K. If /K is totally ramified, then r = 1, dim(r /M ) = [ : K], c 2 (R /M, K) = t+1, and c [:K](R /M, K) = ord K(disc(/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 c(t, K) to c(t, K) and 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 /K, K) s for the Weil restriction R /K, where is a totally ramified separable extension of K. In Thm. 4.4 we compute the first invariant c 1 (R /K, K) for the norm-one subtorus R /K := ker ( ) Nm /K : R /K G m in R /K (G m ), where /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 /K, K), i 2 for the norm-one torus R /K. For a continuous Galois representation ρ : Gal(K sep /K) G(Z (p) ), we do not know to what extent c(ρ) is related to 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. et 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 of K, denote by O, κ, p, π the ring of integers of, the residue field of O, the maximal ideal of O, and a generator of π respectively. The. et O sh be the strict henselization of O. 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 ramification index is defined by π O = π 2

3 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 et T be a torus over K. We have two notions of Néron models of lft NR T. The lft Néron model T as defined in [BR, Ch. 10], is a smooth group model of T locally of finite type over O, and satisfies T lft NR (O sh ) = T (K sh ). We also have the open subgroup T ft NR of T lft NR such that T ft NR (O sh ) is the maximal bounded subgroup of T (K sh ); T lft NR is of finite type over O. We abbreviate T ft NR to T NR. (2.3) Definition et O be a discrete valuation ring with fraction field. et N 1 N 2 be two free O -modules of finite rank d. et h : N 1 N 2 be an injective O -homomorphism. et 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 m), where Λj (h) : Λ j O (N 1 ) Λ j O (N 2 ) is the j- th exterior power of h. In other words, if π e 1 πe 2... πe dim(v ) are the elementary divisors of the O -module homomorphism h, then α j (h; N 1, N 2 ) = e 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 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 n. (2.4) Definition et T be a torus over K, and let be a finite Galois extension of K such that T is split over. et can T,/K : T NR Spec O Spec O T NR be the unique homomorphism which extends the identity map between the generic fibers, where T := T Spec K Spec. et ( can T,/K ) : ie(t NR ) O O ie(t NR ) be the homomorphism between the ie algebras induced by can T,/K. Define by c 1 (T, K) + + c i (T, K) = c(t, K) = (c 1 (T, K),..., c dim(t ) (T, K)) Q d 0 1 α (( ) i cant,/k ; ie(t NR NR ) O O, ie(t ) ), for i = 1,..., dim(t ). In other words, c 1 (T, K) c dim(t ) (T, K), and π c 1(T,K),..., π c dim(t )(T,K) are the elementary divisors of the O -module homomorphism ( can T,/K. The above definition first appeared in [CYdS, 12.2]; it does note depend on the choice of. 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. 3

4 (2.5) et be a finite separable extension of K. Denote by R /K the torus Res /K (G m ), such R /K (A) = (A K ) NR for every K-algebra A. Its Néron model R /K is equal to Res O /O(G m ), the Weil restriction of G m from O to O. There are natural homomorphisms w : G m R /K and Nm /K : R /K G m. On the group of K-rational points they induce the inclusion K and the /K-norm Nm /K : K respectively. et R /K = ker(nm /K : R /K G m ), R /K = R /K /w(g m ). The character group of X (R /K ) is a free abelian group with basis elements {χ σ } Hom(,K sep ); the Galois group Gal(K sep /K) operates transitively on the χ σ s. The restriction of the character χ σ : R /K G m to = R /K (K) is equal to σ. The character group X (R /K ) of R /K is the subgroup of X (R /K ) generated by elements of the form χ σ χ σ0, where σ 0 is a fixed element of Hom(, K sep ). The character group of R /K is the quotient of X (R /K ) by the subgroup Z ( σ Hom(,K sep ) χ σ). (2.5.1) et be a finite Galois extension of K such that the κ /κ is separable. et Γ = Gal(/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 /p m+1. The function i Γ = i /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 The basic estimates (3.1) Theorem et 1 T T T 1 be a short exact sequence of tori over K. (i) For all 1 k dim(t ) we have c j (T, K) 1 j k 1 j k c j (T, K). (ii) Assume that the residue field κ of O is perfect. Then c j (T, K) c j (T, K) for all 1 i dim(t ). i+dim(t ) j dim(t ) i j dim(t ) 4

5 Proof. Consider the commutative diagram ie(t NR ) O O ie(t NR ) O O ie(t NR ) O O can T,/K can T,/K can T,/K 0 ie(t NR ) NR ie(t ) ie(t NR ) 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 M O O M O O M O O 0 M M M 0. Insert an exact middle row to obtain an enlarged commutative diagram M O O M O O M O O 0 M = M O O M 0 0 M M M 0 where M = (M O O ) (M O O ), and M is equal to the image of M O O in M O O. Notice that all vertical arrows are injective homomorphisms between finite free O -modules of the same rank. The second and the third row are exact, while the first row is a complex of O -modules. Denote by r (resp. s) the rank of the free O-module M (resp. M ), so that M has rank r + s. et k be a natural number such that 0 k r = dim(t ). Then the natural map Λ k O ( M ) Λ k O (M O O ) is a split injection. Therefore α k ( M, M ) α k(m O O, M ). Combined with the trivial estimate α k (M O O, M ) α k( M, M ), we get α k(m O O, M ) α k(m O O, M ). Dividing the above inequality by finishes the proof of 3.1 (i). The proof of 3.1 (ii) is somewhat long and will be divided into several emmas for clarity. (3.1.1) emma et r = rank(m ) = dim(t ), s = rank(m ) = dim(t ) as above. Then α r (M O O, M ) = α s ( M, M O O ) 5

6 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, M ) + α s (M O O, M ) = α r+s (M O O, M ) On the other hand the exactness of the second and the third row in the digram (*) yields The emma follows. α r+s (M O O, M ) = α r ( M, M ) + α s ( M, M ). (3.1.2) emma For i = 1,..., s = rank(m ), we have α i ( M, M ) α i (M O O, M ) + α s ( M, M O O ) 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 -lattices in an s-dimensional -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 -basis w 1,..., w s of N 1 such that π e 1 w 1,..., π es w s is an O -basis of N 0. for suitable natural numbers e 1,..., e s. et v 1,..., v s be an O -basis of N 2. Write w k = s l=1 a kl v l for k = 1,..., s, so we have π e k w k = s l=1 πe k a kl v l for each k. By definition, the determinant of every i i-minor of the s s-matrix B = (π e k a kl) M s (O ) is divisible by π α i(n 0,N 2 ). Hence the determinant of every i i-minor of the s s- matrix A = (a kl ) M s (O ) is divisible by π α i(n 0,N 2 ) s k=1 e k. This means that s α i (N 0, N 2 ) α s (N 0, N 1 ) = α i (N 0, N 2 ) 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 ) = s k=1 e k in the claim by ( ) e k in the claim. max I {1,...,s}, Card(I)=i (3.1.3) emma Notation as above, in particular r = rank(m ). Then for i = 1,..., s = rank(m ). k I k=1 α r+i (M O O, M ) α r ( M, M ) + α i ( M, M ) Proof. This is a consequence of the fact that the natural map is a split injection. Λ r O ( M ) O Λ i O ( M ) Λ r+i O (M O O ) 6

7 Proof of 3.1 (ii), continued. et i be a natural number such that 0 i s = dim(t ). Then α r+i (M O O, M ) α r ( M, M ) + α i( M, M ) by emma = α r (M O O, M ) α r(m O O, M ) + α i ( M, M ) = α r (M O O, M ) α s( M, M O O ) + α i ( M, M ) by emma α r (M O O, M ) + α i(m O O, M ) by emma Dividing the above inequality by gives c j (T, K) c(t K) + 1 j dim(t )+i 1 j i c j (T, K) Since c(t, K) = c(t, K) + c(t, K), the above inequality is equivalent to c j (T, K) c j (T, K). dim(t )+i+1 j dim(t ) We have finished the proof of Thm. 3.1 (ii). i+1 j dim(t ) (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 et 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 is a short exact sequence of O-modules. 0 ie(t NR ) ie(t NR ) ie(t NR) 0 (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 et 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 T NR group over κ. is a unipotent commutative algebraic (ii) We have c i (T, K) > 0 for i = 1,..., dim(t ). 7

8 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 ), and ord h x : X (T ) Z must be trivial because the Γ-coinvariant of X (T ) is finite. Therefore T NR = T lft NR. et T NR κ be the neutral component of 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 T NR κ = U W, where U is a commutative unipotent group over κ and W is a torus over κ. et 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 ] T NR is étale, and [ln ] κ W : W W is an étale isogeny. The kernel cokernel long exact sequence gives an isomorphism T NR κ[l n ] W [l n ] between the two finite étale group schemes over κ. Since [l n ] T NR : T NR T NR is étale and O is strictly henselian, every element of T NR κ[l n ] can be uniquely lifted to an element of 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 ie algebras ( can T,/K induced by the homomorphism ) can T,/K : T NR NR Spec O Spec O T. Its reduction modulo p is canonically identified with the tangent map of the restriction of can T,/K to the closed fibers. The neutral component of the closed fiber of T NR NR is unipotent by (i), and the closed fiber of T is a split torus by definition. So can T,/K is trivial on the neutral component of T NR Spec O Spec O, and its tangent map is trivial. This proves (ii). (3.3.1) Remark (i) Here is an alternative proof of (ii) Suppose that is a finite Galois extension of K such that T is split over K. We may and do assume that is totally ramified over K, in the sense that κ is a purely inseparable extension of κ. By Prop. 3.4 (ii), it suffices to verify the statement of 3.3 for T = R /K. According to the description of ie(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 [[x]] such that Nm /K (u(x)) = 1, then a 1 p. et b n be the image of a n in κ, and let v(x) = 1+ n 1 b n x n be the image of u(x) in κ [[x]]. We have 1 = Nm κ /κ(v(x)) = v(x) [:K]. Hence v(x) = 1. In particular a 1 p. (ii) If κ is separable over κ, the same line of argument provides an explicit lower bound on c 1 (T, K); see Cor (3.4) Proposition et T be a torus over K, and let 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 /K) m T 1 8

9 (ii) There exists a short exact sequence of tori of the form 1 (R /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(/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 m σ) X (T ) σ Γ 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 et be a totally ramified finite Galois extension of K such that κ = κ. et Γ = Gal(/K). Then for each k = 1,...,, we have 1 c 1 (R /K, K) + + c k (R /K, K) = min i Γ (σ τ 1 ). I Γ, Card(I)=k 2 In particular, c 1 (R /K, K) = 0, and i=1 c i (R /K, K) = 1 2 ord K(disc(/K)). Proof. et Γ = Gal(/K) = {1 = σ 1,..., σ n }, n = [ : K] =. et π be a generator of p. As an O-algebra, O is generated by π. Hence i Γ (σ i σ 1 j ) = ord (σ i (π ) σ j (π )) for any 1 i j n. et A be the n n matrix whose (i, j)-th entry is σ j (π i 1 ) for i, j = 1,..., n. For a natural number k, 1 k n, n (c 1 (R /K, K) + + c k (R /K, K)) is equal to the minimum among the orders (measured by ord ) of the determinant of all k k-minors of A; we have to show that it is equal to ord (σ i (π ) σ j (π )). That each sum i<j i,j I min I {1,...,n}, Card(I)=k i<j i,j I σ,τ Γ σ τ ord (π i πj ) 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 emma finishes the proof. 9

10 (4.1.1) emma et 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 a i j 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 Clearly det(b) is divisible by (x i x j ) if i j. The emma follows because Z[x 1,..., x m ] is a unique factorization domain. This finishes the proof of emma and Prop (4.2) Corollary Notation as above, and we assume that K. (i) et t be the natural number such that Γ = Γ t, Γ t Γ t+1. Then for i = 1,..., [Γ : Γ t+1 ] 1. c i (R /K, K) = c i+1 (R /K, K) = i(t + 1) (ii) We have c 1 (R /K, K) = c (R /K, K) = ord K(disc(/K)). More generally, c i (R /K, K) = c i+1 (R /K, K) = ord K(disc(/K)) for i = 1,..., [Γ : Γ t+1 ]. (i 1)(t + 1) Proof. (i) et 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 /K, K) c k (R /K, K) is reached with I being any subset of {σ 1,..., σ a } with k elements; it is equal to k(k 1)(t+1) 2. Hence c k (R /K, K) = (k 1)(t+1) for k = 1,..., a. (ii) For i = 1,..., [Γ : Γ t+1 ], the minimum in the formula in 4.1 for c 1 (R /K, K) c i (R /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 The statement (ii) follows. c i+1 (R /K, K) c (R /K, K) = i 1 σ Γ i Γ(σ) 1 = i ord K(disc(/K)) i(i 1)(t+1) 2. 1 k<l i i Γ(σ k σ 1 l ) (4.3) Corollary Assume that the residue field κ of O is perfect. et T be a torus over K which splits over a finite Galois extension of K. Then c dim(t ) (T, K) ord K(disc(/K)) Proof. This follows from Prop. 3.2, Prop. 3.4 (i), Thm. 3.1 (ii) and Cor. 4.2 (ii). 10.

11 (4.4) Theorem et be a totally ramified finite Galois extension of K such that κ = κ. et Γ = Gal(/K), and let Γ = Γ 0 Γ 1 be the lower-numbering filtration of Γ. et t be the natural number such that Γ = Γ t Γ t+1. Then c 1 (R t+1 /K, K) =. Proof. By Prop. 3.4 (ii), Thm. 3.1 (i) and Cor. 4.2 (i), we have c 1 (R /K, K) c 1 (R /K, K) = t + 1. We will prove the inequality c 1 (R t+1 /K, 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 According to the description of ie(t NR ) in [ds, A1.7], the ie algebra of R /K NR can be identified with all elements a 1 O such that there exists a formal power series of the form u(x) = 1 + a 1 x + n 2 a n x n O [[x]] such that Nm /K (u(x)) = 1. We claim that a n p t+1 for all n 1 for any u(x) as above. Of course this claim implies that the inequality c 1 (R t+1 /K, 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 emma We introduce some notation. et U,0 be the subgroup of O [[x]] consisting of all elements of the form u(x) = 1 + m 1 a m x m, a m O for all m 1. For each integer b 1, denote by U,b the subgroup of U,0 consisting of all elements of the form u(x) = 1 + m 1 a m x m such that a m p b for all m 1. The quotient U,0/U,1 is canonically isomorphic to the subgroup 1 + x κ[[x]] of κ[[x]]. Choose a generator π of p, we obtain an isomorphism β b,π : U,b /U,b+1 x κ[[x]] for each b 1. This map β b,π sends an element u(x) = 1 + π b m 1 a m x m of U,b to the element m 1 a mx m of x κ[[x]], where a m is the image of a m O in κ. (4.4.1) emma Suppose that Γ = Gal(/K) is cyclic of order l, where l is a prime number, and κ = κ. et 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 n + (t + 1)(l 1) Tr /K (π n xo [[x]]) = πk r xo [[x]] where r =. l In particular, Tr /K (p n x O [[x]]) p n+1 x O[[x]] for n = 1,..., t 1 Tr /K (p t x O[[x]]) = pt x O[[x]] Proof. This follows from [S2, V 3, emma 3]. 11

12 (4.4.2) emma Notation as in Suppose that ξ(x) p n x O [[x]], n 1. Then Nm /K (1 + ξ(x)) 1 + Tr /K (ξ(x)) + Nm /K (ξ(x)) Proof. The argument in [S2, V 3 emma 5] works here as well. (4.4.3) emma Notation as in (i) We have Nm /K (U,n ) U K,n for each 0 n t + 1. (mod Tr /K (p 2n )). (ii) et u(x) be an element of U,0. et u(x) be the image of u(x) in 1+x κ[[x]] = U,0 /U,1. Then the image of Nm /K (u(x)) in 1 + x κ[[x]] = U K,0 /U K,1 is u(x) l. (iii) et u(x) be an element of U,n, 1 n t 1. et ξ(x) = β n,π (u(x)) x κ[[x]]. Then the image of Nm /K (u(x)) in x κ[[x]] under β n,πk is α n ξ(x) p, where α n denotes the image of Nm /K(π n) O πk n in κ. (iv) The map U,t /U,t+1 U K,t /U K,t+1 induced by Nm /K can be described as follows. Identify U,t /U,t+1 with x κ[[x]] via β t,π, and identify U K,t /U K,t+1 with x κ[[x]] via β t,πk. Then there exists α, γ κ such that Nm /K sends ξ(x) x κ[[x]] = U,t /U,t+1 to α ξ(x) p + γ ξ(x). Proof. Use 4.4.2, and the argument of [S2] V 3, Prop. 4 and Prop. 5. (4.4.4) emma et be a finite Galois extension of K with group Γ, and assume that κ = κ. Suppose that Γ = Γ t Γ t+1. (i) The /K-norm satisfies Nm /K (U,n ) U K,n for 0 n t + 1, and induces a homomorphism Nm /K,n : U,n /U,n+1 U K,n /U K,n+1 for n = 0, 1,..., t. (ii) Suppose that 1 n t. Identify U,n /U,n+1 with x κ[[x]] via β n,π, and identify U K,n /U K,n+1 with x κ[[x]] via β n,πk. Then Nm /K,n : x κ[[x]] U,n /U,n+1 U K,n /U K,n+1 β n,π 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 /K,n : ξ(x) δ n ξ(x) [:K] ξ(x) x κ[[x]]. (iii) Notation as in (ii). For n = t, there exists an additive polynomial P (Y ) κ[y ] of degree [ : K] and separable degree [Γ : Γ t+1 ] such that Nm /K,t : ξ(x) P (ξ(x)) 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 /K, starting from K, by a sequence of successive cyclic extensions. Apply emma and use induction. 12

13 (4.4.5) emma Notation as in Suppose that u(x) is an element of U,0 such that Nm /K (u(x)) = 1. Then u(x) U,t+1. Proof of and Thm As explained at the beginning of the proof of 4.4, emma implies Thm We have seen in that u(x) U,1. Apply emma (ii) repeatedly, we deduce that u(x) U,t. It remains to show that u(x) U,t+1. et ξ(x) = β t,π. Apply emma (iii) and use the notation there, we have P (ξ(x)) = 0 in κ[[x]], where ξ(x) x κ[[x]] is the image of u(x) under β t,π : U,t /U,t+1 x κ[[x]]. We want to deduce, from the equation P (ξ(x)) = 0, the desired conclusion that 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 pk + + 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 (ξ(x) Card(Γ t+1) ) = P (ξ(x)) = 0. Therefore ξ(x) Card(Γ t+1 ) = 0 by Hensel s emma. Hence ξ(x) = 0. (4.5) Corollary et be a finite Galois extension of K such that κ /κ K is separable. et T be a torus over K which is split over. Assume that T is anisotropic over the maximal unramified extension of K. Then c 1 (T, K) t+1, 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 Examples (5.1) Example et /K be a totally ramified extension such that Γ = Gal(/K) is cyclic of order p. Assume that κ = 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) et t be the integer such that Γ = Γ t, Γ t+1 = {1}. et T be an anisotropic torus over K of dimension p 1. Then c i (T, K) = 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]/(xp 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. 13

14 (ii) By (i) we may and do assume that T = R /K, so we can apply 4.2 (i). (5.2) Example et m 2 be a natural number. et K = Q 2 (µ 2 m), M = Q 2 (µ 2 m+1), and = Q 1 (µ 2 m+2). et T = R M/K, T = R /K, and let T be the quotient torus T/T. Then c(t, K) = (0, 2 m 1 ), c(t, K) = (2 m 1, 2 m ), c(t, K) = (0, 2 m 2, 3 2 m 2, 2 m ). Proof. The Galois group Γ = Gal(/K) is cyclic of order 4, canonically isomorphic to the subgroup of Z/2 m+2 Z) generated by the image of m ; denote this element by σ. et ζ be a primitive 2 m+2 -th root of unity. Then i Γ (σ) = ord (σ(ζ) ζ) = ord (ζ ( 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. et = Gal(M/K), generated by the image σ of σ in. Then and c 1 (T, K) = 2 m 1 by 4.2 (i). ı ( σ) = ord M ( σ(ζ 2 ) ζ 2 ) = ord M (ζ 2 ( 2)) = 2 m We still have to compute c(t, K). By [Ch1] 4.5 and 4.8, we have a natural short exact sequence 0 ie(t NR ) ie(t NR ) ie(t NR) 0 which is compatible with the map on ie algebras induced by the base change map can /K. We identify ie(t NR ) with O M, and ie(t NR ) with O. So ie(t NR ) is identified with O /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 ie(t NR ) with O O. Hence the map can T,/K : ie(t NR ) O O ie(t NR ) is identified with the map from (O /O M ) O O O O which sends each element ȳ O /O M represented by y O to the element (σ 2 (y) y, σ 3 (y) σ(y)) O O. Using the O-basis { ζ, ζ 3 } of O /O M, one finds that the elementary divisors of can T,/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. 14

15 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. et O K be the (2)-adic completion of Z[x, y], O M1 be the (2)-adic completion of Z[u, y], O M2 be the (2)-adic completion of Z[x, v], O M3 be the (2)-adic completion of Z[x, uv], and let O 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 (resp. O M1, O M2, O M3, 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, the fraction field of O K, O M1, O M2, O M3 respectively. et T = R /K, and let T i = R Mi /K, T i = T/T i, i = 1, 2, 3. Then c(t, K) = (0, 1, 1, 2), c(t i, K) = (0, 1), 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(/K) is isomorphic to (Z/2Z) 2. et σ (resp. τ) be the element of Gal(/K) such that σ(u) = v, σ(v) = v (resp. τ(u) = u, τ(v) = v), so Gal(, K) = {1, σ, τ, στ}. The set {1, u, v, uv} is an O K -basis of O. So the c i (T, K) s are the elementary divisors of the matrix u u u u v v v v uv uv uv uv A simple computation shows that c(t, K) = (0, 1, 1, 2). A similar but simpler calculation shows that c(t i, K) = (0, 1), i = 1, 2, 3. The ie algebra of T i NR is naturally isomorphic to the quotient of the ie algebra of T NR NR by the ie algebra of T i 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 c(t i, K) = (1, 2) because there is no other possibility. (5.4) Example We generalize 5.3 to odd primes. et 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. et O K and O be the (2)-adic completion of Z[x, y] and Z[x, y] respectively; denote the fraction fields by K,. The field is separable over K of degree p 2. et K 1 = K(µ p ), 1 = (µ p ). Then 1 /K 1 is Galois with Gal( 1 /K 1 ) = (Z/pZ) 2. et T = R /K, T 1 = R 1 /K 1. Then each c i (T, K) has the form i, where i is an integer, 0 i 2p 2. For a natural number i p 1 i with 0 i 2p 2, the multiplicity of in c(t, K) is equal to min(i + 1, 2p 1 i). p 1 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 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(/k) = (p) 2p2, while disc( 1 /K 1 ) = (p) 2p2 = (ζ p 1) 2(p 1)p2. Notice that c p 2(T 1, K 1 ) = 2(p 1), and it coincides with the formula 15.

16 in 4.2 (ii) for c dim(t ) (T, K) for an induced torus T, if we interpret in 4.2 as [ : K] when κ /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 p 1,..., u p m]: Then disc( 1 /K 1 ) = n(p 1)p n, and c p n(r 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 [Beg]. Bégueri: Dualité sur un corps local à corps résiduel algébriquementclos. Mémoire Soc. Math. de France, supplément 108, [BR] [BX] S. Bosch, W. ütkebohmert, and M. Raynaud. Néron Models, volume 21 of Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge. Springer-Verlag, S. Bosch, X. Xarles. Component groups of Néron models via rigid uniformization. Math. Ann., 306: , [CYdS] C.-. Chai and J.-K. Yu. Congruences of Néron models for tori and the Artin conductor. With an appendix Another approach to the characteristic p case by E. de Shalit. Accepted for publication by Ann. of Math. [Ch1] [Ch2] [D] [ds] [El] C.-. Chai. Néron models for semiabelian varieties: congruence and change of base field. Asian J. Math., 4: , C.-. Chai. The loss of Faltings height due to stabilization is measured by a bisection of the Artin conductor. Preprint, P. Deligne. es corps locaux de caractéristique p, limites de corps locaux de caractéristique 0. In the monograph Représentations des groupes réductifs sur un corps local by Deligne, Bernstein and Vigneras, pages Hermann, E. de Shalit. Another approach to the characteristic p case. Accepted for publication by Ann. of Math. R. Elkik. Solution d équations a coefficients dans un anneau hensélien. Ann. Sci. Éc. Norm. Sup., 6: , [S1] J.-P. Serre. Groupes proalgébriques. Publ. Math. I.H.E.S., 7:4 68, [S2] J.-P. Serre. Corps ocaux, 3rd corrected ed. Hermann, [S3] J.-P. Serre. Galois Cohomology. Springer,

17 Ching-i Chai Department of Mathematics University of Pennsylvania Philadelphia, PA U. S. A. home page: chai 17