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Néron models for semiabelian varieties: congruence and change of base field Ching-Li Chai 1 Version 1.3, August 10, Introduction Let O be a henselian discrete valuation ring with perfect residue

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Néron models for semiabelian varieties: congruence and change of base field Ching-Li Chai 1 Version 1.3, August 10, Introduction Let O be a henselian discrete valuation ring with perfect residue field. Denote by K the fraction field of O = O K, and by p = p K the maximal ideal of O. Then every abelian variety A over K has a Néron model A over O. The Néron model A of A is a smooth group scheme of finite type over O, characterized by the property that for every finite unramified extension L of K, every L-valued point of A K extends uniquely to an O L -valued point of A. We refer to the book [BLR] for a thorough exposition of the construction and basic properties of Néron models. In general, the formation of Néron models does not commute with base change. Rather, for every finite extension field M of K, we have a canonical homomorphism can A,M/K : A Spec OK Spec O M A M from the base change of the Néron model to the Néron model of the base change, which extends the natural isomorphism between the generic fibers. If A has semistable reduction over O K, i.e. if the neutral component of A is a semiabelian scheme, then can A,M/K is an open immersion. We define a numerical invariant c(a, K) of A as follows. Let L/K be a finite separable extension such that the abelian variety A has semistable reduction over O L. Let e(l/k) be the ramification index of L/K. Define ( ) 1 c(a, K) := e(l/k) length Lie A L O L can A,L/K (Lie A O L ) Notice that c(a, K) does not depend on the choice of L/K. The invariant c(a, K) measures the failure of A to have semistable reduction over O K ; it is equal to zero if and only if A has semistable reduction over O K. Thus c(a, K) may be regarded as a sort of conductor of A. We will call it the base change conductor of A. One motivation of this paper is to study the properties of the invariant c(a, K) and determine whether it can be expressed in terms of more familiar ones, for instance the Artin conductor or the Swan conductor of the l-adic Tate module attached to A. 1 partially supported by grant DMS from the National Science Foundation 1 In a similar fashion one can attach to each semiabelian variety G over K a non-negative rational number c(g, K); see 2.4 for the precise definition. For a torus T over K it has been shown, by E. de Shalit and independently by J.-K. Yu and the author, that c(t, K) = 1 2 a (X (T ) Z Q), ( ) one-half of the Artin conductor of the linear representation of Gal(K sep /K) on the character group of T ; see [CYdS]. This answers, in the affirmative, a question posed by B. Gross and G. Prasad. The main geometric result of [CYdS] says that the congruence class of the Néron model T of T is determined by the congruence class, with perhaps a higher congruence level, of the Galois twisting data of T. More precisely, for a given torus T there exists an integer m such that for any N 1 the congruence class of the Néron model T of T modulo p N K is determined up to unique isomorphism, by the congruence class modulo p N+m K in the sense of [D], of the quadruple (O K, O L, Gal(L/K), X (T )). Here L is a finite Galois extension of K which splits T, and X (T ) is the character group of T with natural action by the Galois group Gal(L/K). With this congruence result at our disposal, the calculation of the base change conductor c(t, K) in [CYdS] proceeds in several steps. For an induced torus of the form, T = Res M/K G m, where M is a finite separable extension of K, an easy computation shows that c(t, K) is equal to half of the exponent of the discriminant disc(m/k). Therefore the formula (*) holds for products of induced tori. Suppose that the char(k) = 0. One shows, as a consequence of Tate s Euler-Poincaré characteristic formula, that c(t, K) depends only on the K-isogeny class of T. Therefore the base change conductor is an additive function on the Grothendieck group of finite dimensional Q-rational representations of Gal(K sep /K). According to Artin s theorem on the characters of a finite group we know that the K-isogeny class of T is a Q-linear combination of the isogeny classes of induced tori of the form Res M/K G m, where M runs through subextensions of L/K. So the formula (*) holds in general. When char(k) = p 0, one approximates (O K, O L, Gal(L/K), X (T )) by a quadruple (O K0, O L0, Gal(L 0 /K 0 ), X (T 0 )) with char(k 0 ) = 0 as in [D]. One concludes by the geometric result on congruence of Néron models that the formula (*) and the isogeny invariance of c(t, K) still hold. Here what makes it possible to approximate tori over local fields of characteristic p by tori over local fields of characteristic 0 is the following, often under-appreciated, fact: The group of automorphisms of a split torus T of dimension d is isomorphic to GL d (Z), and two split tori T 1, T 2 of the same dimension over local fields K 1, K 2 are congruent if O K1, O K2 are congruent. In this article we examine how far the method used in [CYdS] can be generalized to the case of abelian varieties, or more generally semiabelian varieties. Happily the geometric 2 result on congruence generalizes to Néron models for abelian varieties as expected: Given an abelian variety A over K, there exists a constant m 0 such that for each N 0, A Spec O Spec (O/p N ) depends only on the (mod p N+m )-congruence class of the Galois action on the degeneration data for A; see Theorem 7.6 for the precise statement and 7.1, 7.3 for the relevant definitions. We make a digression in this paragraph to explain the above congruence statement. Suppose that L/K is a finite Galois extension of K such that A L has semistable reduction over O L. Then the neutral component of the Néron model of A L can be constructed, by a uniformization procedure, as a quotient of a semiabelian scheme G over O L by a subgroup Y of periods; see [FC, chap. 2, 3]. The Galois L/K-descent data for A K induces a semi-linear action of Gal(L/K) on ( G, Y ). There is a natural notion of congruence between two such degeneration data ( G 1, Y 1 ), ( G 2, Y 2 ) with actions by Gal(L 1 /K 1 ), Gal(L 2 /K 2 ) respectively, provided that the two sextuples (O K1, p K1, O L1, p L1, Gal(L 1 /K 1 ), Y 1 ) and (O K2, p K2, O L2, p L2, Gal(L 2 /K 2 ), Y 2 ) are congruent modulo a given level, say m + N. Our congruence result says that congruence between two degeneration data as above implies congruence between the Néron models A K1 and A K2. We illustrate the notion of congruence of degeneration data in the case of Tate curves. Suppose we are given an isomorphism α between and (O K1, p K1, O L1, p L1, Gal(L 1 /K 1 ), Y 1 ) mod p m+n K 1 (O K2, p K2, O L2, p L2, Gal(L 2 /K 2 ), Y 2 ) mod p m+n K 2. After picking a generator for Y 1 and a corresponding generator for Y 2 under α, we obtain periods q i p Li, i = 1, 2. Then we say that two degeneration data (G m, q1 Z ), (G m, q2 Z ) are congruent at level m + N if q 1 and q 2 have the same order a and their respective classes in p a L 1 (O L1 /p m+n K 1 ) = p a L 2 (O L2 /p m+n K 2 ) correspond under α. However unlike the case of tori, one cannot deduce the validity of a general statement about the base change conductor for abelian varieties over local fields by reducing to characteristic 0 using the congruence result explained above. The difficulty is that an abelian scheme A OK over O K in positive characteristic may have too many automorphisms, such that no matter how one approximates the moduli point of A OK by O K -valued points with char(k ) = 0, the resulting abelian scheme A over O K will have too small a group of automorphisms, causing it impossible to approximate some Galois twist of A by abelian varieties over local fields of characteristic 0. In some situations the base change conductor c(a, K) has a simple expression. For instance if the abelian variety A has potentially totally multiplicative degeneration, i.e. there exists a finite extension L/K such that the neutral component of the closed fiber of the Néron model A L of A L is a torus, then c(a, K) is equal to a quarter of the Artin 3 conductor of the l-adic Tate module V l (A) of A for a prime number l which is invertible in the residue field κ. See Cor. 5.2, and also see Prop. 7.8 for a more general result. Suppose either that the residue field κ of K is finite, or that char(k) = 0, then the calculation of the base change conductor c(a, K) for a general abelian variety A over K can be reduced to that of an abelian variety B over the completion K of K with potentially good reduction, in the following sense. Recall that c(a, K) = c(a, K) since A K = A Spec O Spec O K. According to the general theory of degeneration of abelian varieties, the abelian variety A K over K can be uniformized as the quotient of a semi-abelian variety G over K by a discrete lattice Y, where G is an extension of an abelian variety B over K by a torus T, and B has potentially good reduction. In the above situation, Theorem 5.3 asserts that c(a, K) = c(t, K) + c(b, K). Despite what the tori case may suggest, in general the base change conductor for abelian varieties over a local field K does change under K-rational isogenies: There exist abelian varieties A 1, A 2 which are isogenous over K, yet c(a 1, K) c(a 2, K). In 6.10 we give two such examples, one with K = Q p and another with K = κ[[t]], where κ is a perfect field of characteristic p 0. In view of these examples, we see that c(a, K) cannot be expressed in terms of invariants attached to the l-adic Tate modules V l (A) of A. There is one positive result in this direction: If A 1, A 2 are abelian varieties over a local field K of characteristic 0 which are K-isogenous, and if there exists a finite separable extension L of K such that the neutral component of the closed fiber of the Néron models of each A i over O L is an extension of an ordinary abelian variety by a torus, then c(a 1, K) = c(a 2, K); see Theorem 6.8. The examples mentioned earlier show that the ordinariness assumption cannot be dropped. It has long been known since the creation of the Néron models that the formation of Néron models does not preserve exactness, nor does it commute with change of base fields. Thus the phenomenon that the base change conductor c(, K) has some nice properties, to the effect that many defects on the level of Néron models themselves often cancel out when measured by c(, K), may be unexpected. This may explain why the invariant c(, K) has attracted little attention before. This article and its predecessor [CYdS] are among the first to study the base change conductor; many basic questions concerning this invariant remain unsettled. A list of open problems, together with some comments, can be found in 8. It is a pleasure to thank J.-K. Yu; this paper could not have existed without him. Indeed he was the first to observe that the Néron models for tori with congruent Galois representations should be congruent, and that this can be brought to bear on the problem of Gross and Prasad on the base change conductor c(t, K) for tori. The author would also like to thank S. Bosch, E. de Shalit, X. Xavier and especially to S. Shatz for discussion and encouragement. Thanks are due to the referee for a very thorough reading of the manuscript. The seed of this work was sowed in the summer of 1999 during a visit to the National Center for Theoretical Science in Hsinchu, Taiwan; its hospitality is gratefully acknowledged. 4 2. Notations (2.1) Let O = O K be a discrete valuation ring with fraction field K and residue field κ. Let p = p K be the maximal ideal of O and let π = π K be a generator of p. The strict henselization (resp. the π-adic completion) of O will be denoted by O sh (resp. Ô). Their fields of fractions will be denoted by K sh and K respectively. The residue field of O sh is κ sep, the separable closure of κ. (2.2) In this paper T (resp. G or G, resp. A or B), sometimes decorated with a subscript, will be the symbol for an algebraic torus (resp. a semiabelian variety, resp. an abelian variety). Often G fits into a short exact sequence 0 T G B, so that the semi-abelian variety G is an extension of the abelian variety B by a torus T, called a Raynaud extension. Such usage conforms with the notation scheme used in [FC, Ch. 2, 3], which we will generally follow. (2.3) It is well-known that semiabelian varieties over K have Néron models. The Néron models come in several flavors. For a be semiabelian variety G over K, we have the lft Néron model G lft as defined in [BLR], the open subgroup scheme G ft of G lft such that G ft (O K ) is the maximal bounded subgroup of G(K), the open subgroup scheme G conn of G ft such that the closed fiber of G conn is the neutral component of G ft. The lft Néron model G lft is a group scheme which is smooth, therefore locally of finite type, over O K ; it satisfies G lft (O sh K ) = G(Ksh ft ). The other two models, G and G conn, are ft smooth and of finite type over O K. We will abbreviate G to G. Clearly A lft = A for every abelian variety A over K. (2.4) In this subsection we define numerical invariants c(g, K) and c(g, K) = (c 1 (G, K),..., c g (G, K)), where g = dim(g), for a semiabelian varietie G over K. For each semiabelian variety G over K there exists a finite extension field L/K such that G L conn is a semiabelian scheme over O L. Actually there exists a finite extension L/K which has the additional property that it is separable, or even Galois, but this is not necessary for the definition. Let can G,L/K : G Spec OK Spec O L G L be the canonical homomorphism which extends the natural isomorphism between the generic fibers. Define non-negative rational numbers 0 c 1 (G, K) c g (G, K), where g = dim(g), by Lie G L can G,L/K (Lie G O L ) = i=1 O L p e(l/k) c i(g,k) L 5 Let c(g, K) = (c 1 (G, K),..., c g (G, K)) and c(g, K) = c 1 (G, K) + + c g (G, K). Notice that c(g, K) and the c i (G, K) s do not depend on the choice of the finite extension L/K such that G has semistable reduction over L. It is easy to see that c(g, K) is equal to zero if and only if G has semistable reduction over K. We call c(g, K) the base change conductor of G, and the c i (G, K) s the elementary divisors of the base change conductor of G. (2.5) Let 0 T G B 0 be a Raynaud extension as above. We denote by X = X (T ) the character group of T ; it is an étale sheaf of free abelian groups of finite rank over Spec K. Therefore one can also think of X as a module for the Galois group Gal(K sep /K). Every Raynaud extension 0 T G B 0 corresponds to a K-rational homomorphism c : X B t, where B t is the dual abelian variety of B. (2.6) Let Y be an étale sheaf of free abelian groups of finite rank over Spec K. A homomorphism ι : Y G from Y to a semi-abelian variety G corresponds to a K-rational homomorphism c t : Y B, together with a trivialization τ : 1 Y X (c t c) P 1 of the biextension (c t c) P 1 over Y X. 3. Uniformization of abelian varieties Throughout this section O is assumed to be a complete discrete valuation ring. Our purpose here is to review the basic facts about uniformizing an abelian variety A over K as the quotient of a semi-abelian variety G over K by a discrete subgroup ι : Y G. Here Y is an étale sheaf of free abelian groups of finite rank over Spec K. The semi-abelian variety G fits into a Raynaud extension 0 T G B 0, such that the abelian variety B has potentially good reduction, and dim(t ) = rank(y ). Moreover the above data satisfies the positivity condition on page 59 of [FC], in the definition of the category DD, after a finite separable base field extension L of K such that T is split over L and B has good reduction over O L. This positivity condition is independent of the finite separable extension L one chooses. For future reference, an embedding ι : Y G which satisfies the above positivity condition will be called a K-rational degeneration data, or a degeneration data rational over K. A K-rational degeneration data as above is said to be split over an extension L of K if Y is constant over K (equivalently, the torus T is split over L) and B has good reduction over L. Every degeneration data rational over K splits over some finite Galois extension of K. (3.1) Proposition (i) Let O = O K be a complete discrete valuation ring. Then every K-rational degeneration data ι : Y G gives rise to an abelian variety A over K such that A is the quotient of G by Y in the rigid analytic category. Conversely every abelian variety A over K arises from a K-rational degeneration data ι : Y G. (ii) Suppose that ι : Y G is the K-rational degeneration data for an abelian variety A over K. Then for every Galois extension L/K of K such that the degeneration data 6 splits over L (equivalently, A has semistable reduction over O L ), there is a natural isomorphism A(K) = ( G(L)/Y (L)) Gal(L/K). This isomorphism is functorial in L/K. Proof. The references for the quotient construction and the uniformization theorem are [M], [R1], [FC, Ch. 2, 3] and [BL]. They are written in the case when the abelian variety A has semistable reduction over O, or equivalently when the degeneration data ι : Y G is split over K. For instance [FC, Prop. 8.1, p. 78] is a reference for (ii) in the case when A has semistable reduction over K. The slightly more generally statement in the proposition follows from the semistable reduction case by descent. 4. The invariant c( G, K) for semiabelian varieties The main result of this section is (4.1) Theorem Assume either that char(k) = 0 and the residue field κ of O = O K is perfect, or that the residue field κ is finite. Then for every Raynaud extension 0 T G B 0 over K, we have c( G, K) = c(t, K) + c(b, K). (4.2) Remark (i) Our proof of Theorem 4.1 in the two cases are quite different technically. When the residue field κ is finite we use the Haar measure on the group of rational points on finite separable extensions of K. This proof is valid when K has characteristic p but we have difficulty translating it to the more general situation when the residue field is perfect but not finite. (ii) The proof for the case when the char(k) = 0 is somewhat indirect. First we establish it in Cor. 4.7 in the case when T is an induced torus; this part is valid for every discrete valuation ring O. Then we show in Lemma 4.9 that the base change conductor c( G, K) stays the same under any K-isogeny whose kernel is contained in the torus part T of G, if char(k) = 0. The proof of Lemma 4.9 is valid only when char(k) = 0. The following lemma may indicate that the statement of Theorem 4.1 is plausible. (4.3) Lemma Assume that the residue field κ of O is algebraically closed and K is complete. (i) For every torus T over K, we have H j (K, T ) = (0) for all j 1. (ii) Let M be a free abelian group of finite rank with a continuous action by Gal(K sep /K). Then H 1 (Gal(K sep /K), M) is finite, while H j (Gal(K sep /K), M) = (0) for all j 3. Proof. This lemma is certainly known. We provide a proof for the readers convenience. According to [S2, Chap. XII], the Brauer group of every finite extension of K is trivial, hence by [S3, Chap. II, 3, Propl 5] the cohomological dimension of Gal(K sep /K) is at most 1. So the strict cohomological dimension of Gal(K sep /K) is at most 2. The statement for j 3 in both (i) and (ii) follows. 7 (i) Let L/K be a finite separable extension which splits T. Let X (T ) be the cocharacter group of T. The induced module Ind Gal(Ksep /K) Gal(K sep /L) X (T ) is the cocharacter group of an induced torus T over K. The natural surjection gives an exact sequence of tori over K. By Shapiro s lemma, Ind Gal(Ksep /K) Gal(K sep /L) X (T ) X (T ) 1 T T T 1 H 2 (K, T ) = X (T ) Z H 2 (L, (L sep ) ) = (0). So H 2 (K, T ) = H 3 (K, T ) = (0), the second equality holds because scd(k) 2. We have shown that H 2 (K, T ) = (0) for every torus T over K, especially H 2 (K, T ) = (0). Again from the long exact sequence, we get H 1 (K, T ) = H 2 (K, T ) = (0). This proves (i). Anot

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