Ching-Li Chai 1 & Frans Oort

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1 1. Introduction CM-lifting of p-divisible groups Ching-Li Chai 1 & Frans Oort Let p be a prime number fixed throughout this article. An abelian variety A over the algebraic closure F of F p is said to admit a CM-lifting to characteristic 0 if there exists a local integral domain R (which can be chosen to be a complete discrete valuation ring) with residue field κ = F and an abelian scheme A over R such that A Spec(R) Spec(κ) is isomorphic to A and End 0 (A) contains a commutative semisimple algebra F over Q with [F : Q] = 2 dim(a). Such an abelian scheme A is said to be of CM-type. The main result of [9] below provides many examples of abelian varieties over F which cannot be CM-lifted to characteristic 0. Theorem B in [9]. Let B be an abelian variety over the algebraic closure F of F p such that End 0 (B) is a field and the p-rank at most dim(b) 2. Then there exists an abelian variety A over F isogenous to B which cannot be CM-lifted to characteristic 0. The statements B1 and B2 below explains the strategy for the proof of Theorem B in [9]; B0 establishes notation for B1. See [9, 3.2] for B0, [9, 3.3 & 4.9] for B, and [9, 4.8 & 4.11] for the key step B2. (In [9] the statement B2 is proved for U = A b with b F, but the argument actually proves B2.) B0. Changing B by a purely inseparable isogeny, we may and do assume that the a-number of B is equal to 2. Suppose that B is defined over k. Fix an embedding α p α p B over k, so that the α p -quotients of B F are parametrized by P 1 (F). Denote by A b the α p -quotient attached to b P 1 (F). After modifying B by a further purely inseparable isogeny and a finite base field extension, we may and do assume that the a-number of A b is 1 for every b / P(k). B1. There exists a positive integer Q such that [M(b): M] Q for every abelian variety U over a finite field M k such that A b Spec(k(b)) Spec(F) = U Spec(M) Spec(F). B2. Let N be a given positive integer not divisible by p. There exists a finite extension k 1 of k such that every abelian variety U over F which is isogenous to B via an isogeny of degree dividing Np and admits a CM-lift to characteristic 0 can be defined over the pro-p extension of k 1. The step B2 provides an upper bound on some field of definition of A b if A b admits a CM-lift, while B1 gives a lower bound for every field of definition of A b. We get the desired examples when the two bounds collide: Suppose that A b can be CM-lifted to characteristic 0, then A b can be defined over a field k 2 k 1 such that [k 2 : k 1 ] p by B2, and [k 2 (b): k 2 ] Q] by B1, therefore [k 1 (b): k 1 ] divides Q p. So A b cannot be CM-lifted to characteristic 0 if [k(b): k] does not divide [k 1 : k] Q p. One can view B2 as an obstruction to the existence of CM liftings (and the only one known): for abelian varieties U isogenous to B via isogenies of bounded degree, the field of moduli of U cannot be too big if U can be CM-lifted to characteristic 0. We will show that the assumption in theorem B that End 0 (B) is a field can also be eliminated, answering Question C in [9] affirmatively. This is achieved by a sort of localization, replacing abelian varieties by p-divisible groups in [9]. The notion of CM-liftings generalize in a obvious way to the context of p-divisible groups; see 2.2. Under the present setup but without the 1 Partially supported by a grant DMS from the National Science Foundation 1

2 assumption that End 0 (B) is a field, the localized results are formulated as C1 and C2 below, with C1 slightly stronger than B1; see 3.4 and 4.2 for their proofs. We also state a corollary C2 of C2, which bears more resemblance to B2. C1. There exists a positive integer Q 2 such that [M(b): M] Q 2 for every p-divisible group Z over a finite field M k with the property that Z Spec(M) Spec(F) = A b [p ] Spec(k(b)) Spec(F). C2. There exists a finite extension E of Q p with the following property: For any finite extension K of K 0 = frac(w (F)) and any p-divisible group X over O K of CM-type whose closed fiber is isogenous to B[p ], there exists a finite extension L of K and a p-divisible group Y over the ring of integers O E of a finite totally ramified extension E of E contained in L such that Y Spec(OE ) Spec(O L ) = X Spec(OK ) Spec(O L ). C2 There exists a finite extension k 2 of k such that every p-divisible group X whose base change to F admits a CM lifting to characteristic 0 can be defined over k 2. Note that B0 immediately generalizes to the situation of a p-divisible group Y over F with a non-trial non-ordinary part different from a G 1,h 1 or a G h 1,1 : there exists a p-divisible group X over a finite field k isogenous to Y over F such that the a-number of Y is 2, all endomorphisms of X F are defined over k, and any α p -quotient Y b of Y b corresponding to an element b P 1 (F) P 1 (k) has a-number 1. Moreover C1 and C2 hold when B[p ] is replaced by a p-divisible group X as above; see 3.4 and 4.2 respectively. So we get an obstruction to the existence of CM-liftings of α p -quotients of X from the field of moduli of such α p -quotients. In particular we have the following analog of Thm. B for p-divisible groups, which follows from C1 and C2. Theorem C. Suppose that Y is a p-divisible group over F whose non-ordinary part is nontrivial and different from a one-dimensional p-divisible formal group or the dual of a onedimensional p-divisible formal group. Then there exists a p-divisible group X over F isogenous to Y which cannot be CM-lifted to characteristic 0. See 4.3 for a more precise version of Thm. C. A corollary of Thm. C, already mentioned, is that the assumption on End 0 (X) can be eliminated from the statement of Thm. B. For an illustration, see the example in Preliminaries. In this paper p is a fixed prime number, and F is an algebraic closure of F p. (2.1) Every p-divisible group Y over F is isomorphic to a product Y et Y mult Y (0,1), where Y et is the maximal etale quotient of Y, Y mult is of multiplicative type, and Y (0,1) is a p-divisible group all of whose slopes are strictly between 0 and 1. (2.2) A p-divisible group Z over a complete noetherian domain R of generic characteristic 0 is said to be of CM-type if there exists a commutative semi-simple algebra F over Q p and an injective homomorphism ι: F End 0 (Z) such that dim Qp (F ) = height(z). In the above situation we say that Z has CM by F, or more precisely by the order F End(Z) of O F, where O F is the ring of integers in F. The CM-type Φ of (Z, E, ι) is the element of character ring of the algebra E afforded by the tangent space Lie(Z) R frac(r) of Z over the fraction field frac(r) of R, considered as a vector space over frac(r) with frac(r)-linear action by F. Let Z be a p-divisible group over a field κ of characteristic p. A p-divisible group Z of CM-type over a complete noetherian local domain (R, m) with generic characteristic 0 is a CM- 2

3 lifting of Z if there exists a ring homomorphism R/m κ such that Z Spec(R) Spec(κ) = Z. If Z admits a CM-lifting, then there exists a finite extension κ 1 of κ such that Z Spec(κ) Spec(κ 1 ) admits a CM-lifting to a complete discrete valuation ring (R, m) of characteristics (0, p). (2.3) Suppose that (F, Φ) is the CM-type of a p-divisible group Z of CM-type over R as in 2.2, and the ring of integers O F operates on Z. Write F as a product F = F 1 F c of fields. Let Z = Z 1 Z c (resp. Z = Z 1 Z c ) be the decomposition of Z (resp. Z) corresponding to be the decomposition of Z corresponding to the product decomposition of E. Notice that each Z i is an isoclinic p-divisible group with height(z i ) = [F i : Q p ]. Then the CM-type Φ of E decomposes into a product of the CM-types (E i, Φ i ) of the Z i s. For each i = 1,..., c, Φ i is the character of the algebra F operating K-linearly on T Zi R K, where K = frac(r). Explicitly, if we fix an algebraic closure K of K, then Φ i can be identified with a subset of Hom(F i, K). Let E i = E(F i, Φ i ) = E( ι Φ i ι(x) : x F i ) be the reflex field of (F i, Φ i ), or equivalently the field of definition of the character Φ i. The reflex field E = E(F, Φ) is by definition the compositum of the reflex fields E i = E(F i, Φ i ), i = 1,..., c. It is clear that the fraction field of R contains the reflex field E(F, Φ). A CM-type (F, Φ) = (F 1, Φ 1 ) (F c, Φ c ) is said to be compatible with F End 0 (Z), or compatible with Z for short, if card(φ i ) = dim(z i ) for each i; equivalently, Z i is isoclinic of slope card(φ i )/[F i : Q p ] for each i. It is easy to see that if Z is a CM-lift over a mixed characteristic local ring (R, m) of a p-divisible group Z over R/m with CM-type (F, Φ), then (F, Φ) is compatible with Z. We collect some well-known facts in the following proposition. (2.4) Proposition (1) Suppose that R is a mixed characteristics complete discrete valuation ring with residue field F, and Y 1, Y 2 are p-divisible groups over R with the same CM-type (F, Φ). Then Y 1 and Y 2 are F -linearly isogenous over R. (2) Let E be a finite extension field of Q p, and let K be a finite extension field of K 0 := frac(w (F)). Let Y 1 be a p-divisible group over O E, Y 2 be a p-divisible group over O K, both with the same CM-type (F, Φ). Then there exists a finite extension E 1 of E contained in K, a finite flat subgroup C Y 1 Spec(OE ) Spec(O E1 ) and an F -linear isomorphism ( Y1 Spec(OE ) Spec(O E1 )/C ) Spec(OE1 ) Spec(O K1 ) where K 1 := E 1 K is the compositum of E 1 and K. Y 2 Spec(OK ) Spec(O K1 ), (3) Let F be a commutative semisimple Q p -algebra and let Φ be a CM-type for F. Then there exists a p-divisible group Y over O K with CM-type (F, Φ), where K = E K 0 and E = E(F, Φ) is the reflex field of (F, Φ). (4) Let Z 1 be a p-divisible group over a finite field k 1. Let F be a maximal commutative semisimple Q p -subalgebra of End 0 (Z). Let Φ be a CM-type of F compatible with Z 1. Let E 2 be a finite extension field of Q p which contains the reflex field E = E(F, Φ) such that the residue field κ 2 of E 2 contains k 1. Then there exists a p-divisible group Z over O E2 of CM-type (F, Φ) such that Z Spec(R) Spec(κ 2 ) is F -linearly isogenous to Z 1 Spec(k1 ) Spec(κ 2 ). 3

4 Sketch of proof. A proof of Prop. 2.4 can be found in the appendix: (1) follows from 5.7 (ii), (2) follows from (1), (3) follows from 5.7 (i) and (4) follows from The appendix in 5 treats complex multiplication of p-divisible groups over complete discrete valuation rings of characteristic (0, p) up to isogeny from the perspective of p-adic Hodge theory. It is also possible to prove 2.4 using the classical theory of complex multiplication of abelian varieties; we sketch the idea. It is known that the Galois representation of a p-divisible group over the ring of integers of a local field is algebraic when restricted to the inertia group; see [10, Chap. 3]. The p-divisible groups Y 1, Y 2 in (1) have the same CM-type, so they have the same Galois representation by [11, 2.3]. We conclude (1) by Tate s theorem on extending homomorphisms between p-divisible groups. To prove the existence statement (3), we may and do assume that F is a field. Choose a CM-field L and a CM-type Φ L such that (F, Φ) is a component of (L, Φ L ) Q p ; the existence of such a pair Φ L follows from Hilbert irreducibility. Let ι: L F alg be a p-adic embedding of L which induces an isomorphism of L F for a place of L above p. Let M F alg be the reflex field of (L, Φ L ) contained in F alg, and let v be the place of M induced by the embedding M F alg, so M v = E. By [2, 5.2], there exists an O L -linear abelian variety A over M with CM-type (L, Φ L ) with good reduction at v. Let A 1 be the abelian scheme over O Mv which extends A Spec(M) Spec(M v ), and let A 2 be the base change to O K of A 1. Then the -divisible group A 2 [ ] of A 2 is a p-divisible group over O K with CM-type (F, Φ), as required in (3). We have said that (2) follows from (1); now we explain that (4) follows from the proof of (3). As before we may and do assume that F is a field. Repeating the argument for (3) in the preceding paragraph. Then Z 3 := A 1 [ ] is a p-divisible group over E of CM-type (F, Φ). Make a suitable unramified K/E 2 -twist of Z 3 Spec(OE ) Spec(O E2 ) to produce a p-divisible group Z over O E2 such that the Frobenius of the closed fiber of Z is equal to the Frobenius of Z 1 Spec(k1 ) Spec(κ 2 ). Then Z has the required properties. (2.5) Proposition Let E = E(F, Φ) be the reflex field of a p-adic CM-type (F, Φ). Let E 1 be a finite extension field of E. Then there exists a p-divisible group Y over O E1 with action by O F of CM-type (F, Φ) such ρ Y (Γ E1 ) = ρ Y (I E1 ) as subgroups of O F, where Γ E 1 is the Galois group Gal(E alg 1 /E 1), Γ IE1 is the inertia subgroup of Γ E1, and ρ Y : Γ E O F is the Galois representation attached to Y. Sketch of proof. See 5.8 for a proof of 2.5. The idea is that p-divisible groups over O E1 of CM-type (F, Φ) correspond to abelian p-adic representations of the abelianized Galois group Γ ab E 1 whose restriction to the inertia subgroup group IE ab 1 Γ ab E 1 is induced by the algebraic homomorphism E 1 F attached to (F, Φ). Since Γ ab E 1 is a direct product of I E1 with Gal(F/κ E1 ), we can adjust a p-divisible group Y 1 over O E1 of CM-type (F, Φ) to make the image of Γ ab E 1 contained in the image of the inertia subgroup. Then the corresponding p- divisible group Y has the required properties. 3. Generic α p -quotients. (3.1) A p-divisible group Y over F is said to be of extended Lubin-Tate type if Y (0,1) is trivial, a one-dimensional p-divisible formal group or the Serre dual of a one-dimensional p-divisible formal group. The property of extended Lubin-Tate type is invariant under isogeny of p- 4

5 divisible groups. Note also that the a-number of any p-divisible group of extended Lubin-Tate type is at most one. (3.2) Proposition Let Y be a p-divisible group which is not of extended Lubin-Tate type. Then there exists a finite field k, a p-divisible group X over k, and a finite extension field k 0 k, with the following properties. (i) X Spec(k) Spec(F) is isogenous to Y, and under the natural map. End(X) End ( X Spec(k) Spec(F) ) (ii) There is an embedding β : α p α p X over k which induces an isomorphism between α p α p and the maximal subgroup of X of α-type. In particular a(x) = 2, and α induces an bijection between P 1 (k ) and subgroups of X Spec(k) Spec(k ) isomorphic to α p over k, where k /k is any subextension field of F/k. (iii) We have a(y b ) = 1 for all b P 1 (F) P 1 (k 0 ), where Y b := (X Spec(k) Spec(F))/β( b ) and b is the subgroup of (α p α p ) Spec(k) Spec(F) isomorphic to α p attached to b. See [9, Prop. 3.2] for a proof of 3.2. Although [9, Prop. 3.2] is formulated for abelian varieties, the proof there applies to p-divisible groups. After extending the base field to k 0, we may and do assume that k 0 = k, i.e. a(y b ) = 1 for all b / P 1 (k). (3.3) Lemma Suppose that b 1, b 2 are elements of P 1 (F) P 1 (k) such that Y b1 = Yb2 as p- divisible groups over F. Then k(b 1 ) = k(b 2 ), where k(b i ) is the residue field of the closed point b i of P 1 over k. Proof. Any isomorphism ψ : Y b1 Yb2 φ: X Spec(k) Spec(F) lifts uniquely to an isomorphism X Spec(k) Spec(F), so φ induces an automorphism of the maximal α p subgroup of X Spec(k) Spec(F) which sends β(a b1 ) to a b2. In particular φ induces an automorphism of P 1 which sends b 1 to b 2. See also the proof of [9, Lemma 3.3]. The following proposition gives a lower bound for any field of definition of an α p -quotient Y b as a p-divisible group. (3.4) Proposition There exists a positive integer Q 2 such that [M(b): M] Q 2 for every p-divisible group Z over a finite field M k with Y b Spec(k(b)) Spec(F) = Z Spec(M) Spec(F). Proof. Let ξ : U Z be an isogeny between p-divisible groups over K such that Ker(ξ) = α p. It is well-known that such an isogeny ξ exists and is unique up to isomorphisms. Moreover U Spec(M) Spec(F) is isomorphic to X Spec(k) Spec(F), i.e. U is a twist of X Spec(k) Spec(M) over the finite field k. Consequently the maximal α-subgroup U[F, V ] of U is a twist of the maximal α-subgroup of X Spec(k) Spec(M) over k. 5

6 The group Aut(X) of all automorphisms of X operates on the maximal α-subgroup α(α p α p ) of X via a finite quotient G 0. Let Q 2 be the largest common multiple of the order of elements of G 0. The assertion [M(b): M] Q 2 follows immediately from the above discussion and the proof of 3.3. (3.5) Remark The proof of 3.4 shows that one can take Q 2 to be the lcm of the order of elements of Im (Aut(X) Aut(X[F, V ])), where X[F, V ] is the maximal α-subgroup of X. So the constant Q 2, which depends only on X, is effectively computable. Although the argument in [9, Prop. 3.2] for the existence of the finite field k 0 in 3.2 (iii) does not provide an algorithm for k 0, in a given situation it is usually not difficult to find a k 0. We work out a simple example below. Suppose that X = G 1,h 1 G h 1,1 over F p h, the product of a one-dimensional formal group and its dual, and k = F p h. Then the finite group Im (Aut(X) Aut(X[F, V ])) is isomorphic to F F, and Q p h p h 2 = p h 1. The product structure of X induces an isomorphism α: α p α p X[F, V ] with α(αp 0) G 1,h 1 and α(0 α p ) G h 1,1. In this case the only b s with α(y b ) = 2 are the two coordinate points (1 : 0) and (0 : 1), so we can take k 0 = F p h = k in 3.2 (iii), 3.3 and An upper bound for the field of definition (4.1) Let h := height(x). By standard number theory, there are only a finite number of commutative semi-simple Q p -algebras of dimension h which can be embedded in End 0 (X). So by Skolem-Noether, there are only a finite number of End 0 (X) -conjugacy classes of triples (F End 0 (X), Φ) where F a maximal commutative semisimple Q p -subalgebra of End 0 (X) and Φ is a CMtype of E compatible with X. Let (F j End 0 (X), Φ j ), j = 1,..., N be a complete set of representatives of conjugacy classes of CM-types which are compatible with X. Let E j = E(F j, Φ j ) be the reflex field of (F j, Φ j ). For each j = 1,..., c, let Z j be an O Fj -linear p-divisible group over the ring of integers O Ej of E j, such that (i) the CM-type of Z j is CM-type (F j, Φ j ), and (ii) ρ Zj (Γ Ej ) = ρ Zj (I Ej ), where ρ Zj : Γ Ej O F j and I Ej is the inertia subgroup of Γ j. is the Galois representation attached to Z j The existence of Z j is guaranteed by 2.4 (3) and 5.8. Denote by κ j the residue field of E j, and let Z j be the closed fiber of Z j. We know by the construction of Z that Z j Spec(κj ) Spec(F) is isogenous to X Spec(k) Spec(F). Suppose that Z is a p-divisible group Z F isogenous to X Spec(κ) Spec(F) which has a CM-lift Z over a finite extension field K of K 0. There exists an integer j with 1 j c such that the CM-type of Z is (F j, Φ j ). By 2.4 (1) and (2), there exist a finite extension field E j of E j, a finite flat subgroup scheme C Z Spec(OEj ) Spec(O E j ) over O E j and an F -linear isomorphism ξ : ( Z j Spec(OEj ) Spec(O E j )/C ) Spec(OE j ) Spec(O L ) Z Spec(OK ) Spec(O L ) 6

7 where L = E j K is the compositum of E j and K. Denote by M the minimal subfield of E j over which the generic fiber C η of C is defined; in other words M is the subfield of E j generated by the residue fields of the maximal points of C. Then C descends ( to a finite flat subgroup ) scheme C Z j Spec(OEj ) Spec(O M ) over O M. Let Z := Z j Spec(OEj ) Spec(O M ) /C, a p-divisible group over O M of CM-type (F j, Φ j ). We know that Z Spec(OM ) Spec(O L ) is isomorphic to Z Spec(OK )Spec(O L ) by construction. In particular the closed fiber Z of Z is a p-divisible group over the residue field κ M of M and Z Spec(κM )Spec(F) is isomorphic to Z Spec(κK )Spec(F), where κ K denotes the residue field of K. As Z varies over all p-divisible groups over F which admit CM-lifts, the fields M also vary, however the residue fields κ M remains constant. In fact the assumption ρ Zj (Γ Ej ) = ρ Zj (I Ej ) implies that M is totally ramified over E j, so κ M = κ j, the residue field of E j. We summarize the above discussion in the following proposition. (4.2) Proposition Notation as above. Let κ be the compositum of the residue field κ j of the reflex fields E j = E(F j, Φ j ), j = 1,..., c. Let Z be a p-divisible group over F isogenous to X Spec(k) Spec(F). If Z admits a CM-lifting to characteristic 0 of CM-type (F j, Φ j ), then there exist a totally ramified finite extension field M of F j and a CM-lifting of Z over O M of CM-type (E j, Φ j ). In particular there exists a p-divisible group Z over κ such that Z Spec(κ) Spec(F) is isomorphic to X Spec(k) Spec(F). Remark. Although in 4.2 we get down to a fixed finite field κ, not the pro-p extension of a finite field as in the statement B2 in the 1, the statement 4.2 is not stronger than B2. Moreover proof of 4.2 does not prove the statement B2 obtained from B2 by deleting the pro-p extension of from the end of B2. (4.3) Theorem Let X be a p-divisible group over a finite field k such that α(x) = 2. Let Q 2 be the integer in 3.4 and let κ be the compositum of the reflex fields κ j as in 4.2. Let b be an element of P(F) such that [kκ(b) : kκ] does not divide Q 2. Then the α p -quotient Y b of X has no CM-lifting to characteristic 0. In particular if X = B[p ] is the p-divisible group attached to an abelian variety B over k, then the α p -quotient A b of B does not admit a CM-lift to characteristic 0. Proof. Immediate from 3.4 and 4.2. (4.4) We illustrate 4.3 in an example. The imaginary quadratic field Q( 7) has class number one. Let p be a prime number which is split in Q( 7), or equivalently p 1, 2, or 4 (mod 7). Let 1 and 2 be the two prime ideals in O Q( 7) above p. Because Q( 7) has class number one, there exists an element π Q( 7) such that πo Q( 7) = Clearly π is a Weil p 3 -number by the product formula. By Honda-Tate there exists a simple abelian variety B over F p 3 with π as its Weil number and O Q( 7) End(B). Then dim(b) = 3, and the p-divisible group decomposes as the product of B[ 1 ] and B[ 2 ], of slope 1/3 and 2/3 respectively. In particular B[F, V ] = α p α p, and we can take k = F p 3 in the statement of 4.3. We note that B is a hypersymmetric abelian variety as defined in [3]; End 0 (B) is a 9-dimensional central division algebra over Q( 7). Every CM-type for compatible with B[ 1 ] has the form (F 1, Φ 1 ) (F 2, Φ 2 ), where F 1 and F 2 are cubic extension fields of Q p, card(φ 1 ) = 1 and card(φ 2 ) = 2. So E(F j, Φ j ) = F j for 7

8 j = 1, 2, and we can take κ = F p 3 in the statement of 4.3. By 3.5, we can take Q 2 = p 3 1 and k = p 3 in the statement of 4.3. Conclusion: the α p -quotient A b of B does not admit a CM-lifting if b / F p 3(p 3 1). 5. Appendix: Crystalline representations of CM type In this section we summarize the local theory of complex multiplication from the perspective of p-adic Hodge theory, and supply proofs of 2.4 and 2.5. Some standard references for p-adic Hodge theory are [5], [6], [7] and [4]. (5.1) Notations. Let K 0 := frac(w (F)). Let L be either a finite extension field of K 0 or a finite extension field of Q p. Denote by Γ L the Galois group Gal(L/L of L, and let I L = Gal(L/L ur ) be the inertial subgroup. (1) Denote by REP? (Γ L ) the Q p -linear -category of a? -representations of Γ L on finite dimensional vector spaces over Q p, where? is one of crys, pcrys, st, pst, dr, HT, and the adjective a? corresponding to its abbreviation? is one of: crystalline, potentially crystalline, semi-stable, potentially semistable, de Rham, Hodge-Tate. Denote by REP ab? (Γ L) -subcategory consisting of all abelian representations in Rep? (Γ L ). (2) Denote by MF f L (φ) (resp. MFf K (φ, N)) the Q p-linear -category of weakly admissible filtered φ-modules (resp. (φ, N)-modules) relative to L. One can regard MF f K (φ) as the -subcategory consisting of objects (M, Fil, φ, N) in MF f L (φ, N) with N = 0. We have inclusions REP crys (Γ L ) REP pcrys (Γ L ) REP pst (Γ L ), REP crys (Γ L ) REP st (Γ L ) REP pst (Γ L ), and REP pst (Γ L ) REP dr (Γ L ) REP HT (Γ L ). The inclusion REP pst (Γ L ) REP dr (Γ L ) is actually an isomorphism, but we will not use this fact: when restricting to abelian local p-adic Galois representations, most of the above inclusions induce isomorphisms. (5.2) Proposition Let L be either a finite extension field of Q p or a finite extension field of K 0. Let ρ: Γ F GL(V ) be an continuous abelian representation of Γ F on a finite dimensional Q p -vector space V. (i) ρ is potentially crystalline ρ is potentially semistable ρ is de Rham ρ is Hodge- Tate ρ is locally algebraic. (ii) ρ is crystalline ρ is semistable ρ IL is algebraic. When L is a finite extension field of Q p, ρ IL is algebraic (resp. locally algebraic) means that there exists a Q p -homomorphism ρ alg : T L := Res L/Qp GL(V ) between algebraic groups over Q p such that ρ rec L and ρ alg coincide on O L (resp. an open subgroup of O L ). Here rec: L Γ L is the (arithmetically normalized) local Artin map. When L is a finite extension of K 0, ρ is algebraic (resp. locally algebraic) means that there is a subfield F K of finite dimension over Q p and an abelian algebraic (resp. locally algebraic) representation ρ 1 of I F such that ρ = ρ 1 ΓK. Remark. Prop. 5.2 is a consequence of [10, Chap. 3, Appendix] and basic facts about the Lubin-Tate formal groups. See also [2, 6.3]. 8

9 Since crystalline representations is most relevant for our purpose, we only state the fundamental result in p-adic Hodge theory to the crystalline case. (5.3) Theorem Let L be either a finite extension field of Q p or K 0. Let L 0 be the maximal absolutely unramified subfield in L. Let B crys and B dr be the Fontaine ring for crystalline and de Rham representations respectively. (i) There is an equivalence of -categories with D crys : REP crys (Γ L ) MF f L (φ) D crys (V ) = (B crys Qp (V )) Γ L, an L 0 -vector space of dimension dim Qp (V ), with the σ-linear action φ induced by the σ-linear action on B crys, endowed with the decreasing filtration Fil on L L0 D crys (V ), defined by Fil i (L L0 D crys (V )) = (L L0 D crys (V )) ( Fil i B dr Qp V ) Γ L. (ii) The functor inverse to D crys is given by V crys (D, φ, Fil) = Fil 0 (B crys K0 D) φ=1. (iii) A crystalline representation (V, ρ) of Γ L comes from a p-divisible group X over O L if and only if the Hodge-Tate weights of (V, ρ) is contained in {0, 1}. If so we have dim Qp V {1} = dim(x ) and dim Qp V {0} = height(x ) dim(x ). Remark. The essential surjectivity of the functor D crys in 5.3 (i) is proved in [4]. The if part of 5.3 (iii) is proved in [1] when p > 2, and in [8] when p = 2. (5.4) Let K be the fraction field of a complete discrete valuation ring O K of characteristics (0, p) whose residue field is algebraically closed. In other words K is a finite extension of K 0. Denote by REP ab crys(γ K ) the Q p -linear tensor category consisting of all abelian crystalline representations of Γ K, with the fiber functor over Q p given by the forgetful functor. Denote by HK ab the Galois group of the neutral Tannakian category REPab crys(γ K ). For any (V, ρ) REP ab crys(γ K ), denote by H(V ) the Galois group of the -subcategory generated by V. Then H(V ) is the smallest algebraic subgroup of GL(V ) defined over Q p which contains ρ(γ K ); see [11, 1]. Since V is Hodge-Tate, we have a Hodge-Tate decomposition C p Qp V = i C p L V {i}, where V {i} is the subset of all elements x C p Qp V such that s(x) = χ cycl (s) i x for all s Γ K. This gradation of C p Qp V defines a cocharacter h V : G m/cp GL(V ) /Cp, which factors through the inclusion H V GL(V ); see [11, 1.4]. Since HK ab is by definition the projective limit of all H V s as V runs through all abelian crystalline representations, the limit of the h V s define a cocharacter h of the pro-algebraic group Hcrys. ab 9

10 For any pair (E, π) consisting of a finite extension field E of Q p and a uniformizer π of E, let f E,π (x) := πx + x q, where q is the cardinality of the residue field of E, and let F E,π (x, y) O E [[x, y]] be the Lubin-Tate formal group law such that f E,π (x) is the endomorphism of F E,π attached to π. Let (V E,π, ρ E,π ) be the p-adic Tate module attached to F E,π tensored with Q p, and let ρ E,π : Γ ab E O E be the associated Galois representation. It is known that ρ E,π rec E (π) = 1, ρ E,π rec E (u) = u 1 u O E, where rec E : E Γ ab E is the arithmetically normalized reciprocity law map. In particular the restriction of ρ E,π to the inertia subgroup I E Γ E is independent of the choice of the uniformizer π E of E. For any subfield E K with [E : Q p ] <, let ρ E K be the restriction to Γ K (V E,π, ρ E,π ). Then the algebraic closure H VE of ρ E (I E ) is T E := Res E/Qp G m. The E-module structure of V E gives a decomposition C p V E = τ ΣE V E,τ, where Σ E is the set of all ring homomorphisms from E to C p. In terms of Hodge-Tate structure of V E, we have C p Qp V E {1} = V E,ι, and C p Qp V E {1} = τ ι V E,τ, where ι: E C p is the inclusion, the composition of E K and K C p. Write h E : G m/cp T E/Cp for the cocharacter of T E corresponding to ι. The pairs (T E, h E ) form a projective system, with transition maps (T E, h E ) (T E, h E ) induced by the norm Nm E /E : T E T E for E E K. (5.5) Theorem Notation as above. (i) The pro-algebraic group HK ab natural homomorphism is an isomorphism. is a connected pro-algebraic torus over Q p. Moreover the K, h) lim (T E, h E ) (H ab E K (ii) Suppose that (V, ρ V ) is an abelian crystalline Q p representation of Γ K such that the corresponding homomorphism HK ab GL(V ) factors through a homomorphism δ : T E GL(V ) for some finite extension E/Q p contained in K. Then the Galois representation ρ V : Γ K GL(V ) is given by ρ V (γ) = δ ( ( rec E O E ) 1(incl(γ)) ) 1 γ Γ K, where incl: Γ K I E is the inclusion map from Γ K to the inertia subgroup I E Γ E, ( ) 1 and rec O is the inverse of the arithmetically normalized local reciprocity law iso- E morphism rec O : O E E I E. (iii) A crystalline abelian p-adic representation (V, ρ V ) of Γ K comes from a p-divisible group over O K if and only if the Hodge-Tate wights of V is a subset of {0, 1}. See [11, 2] for the proof of 5.5 (i). The statement (ii) follows from the corresponding statement for the Tate module attached to the Lubin-Tate formal group laws The statement 5.5 (iii) follows from 5.3 (iii). Implicit in (i) is the fact that the abelian Galois representations V E,π with E K generate REP ab crys(γ K ) as a Q p -linear tensor category. 10

11 (5.6) Example. Suppose that X is a p-divisible group over O K, F is a finite extension field of Q p such that [F : Q p ] = height(x ), and β : F End 0 (X ) is a ring homomorphism. Write V (X ) for the Q p -Tate module of X. The action of F on Lie(X ) OK K gives a subset Φ of Σ F with card(φ) = dim(x ), the CM-type of (X, F, β). The CM-type Φ defines a cocharacter h Φ : G m/cp T F /Cp of T F. Let E be the field of definition of h Φ, or equivalently the field of definition of the character of the K-linear representation of F on Lie(X ) OK K. The p-adic field E, called the reflex field of the CM-type (F, Φ) is a finite extension of Q p contained in K. By the definition of E, there exists an (E, F )-bimodule t such that K E t is isomorphic to Lie(X ) OK K as a (K, F )-bimodule. The homomorphism δ F,Φ : HK ab GL F (V (X )) = T F corresponding to V (X ) can be described in two equivalent ways. (i) the Q p -rational homomorphism whose restriction to E is x det F (x; ) x E, or (ii) the composition T E = Res E/Qp G m Res E/Qp (h Φ ) ResE/Qp (T F ) Nm E/Qp T F. See [11, 2.3] for the first description of δ. The equivalence of (i) and (ii) is well-known in the theory of complex multiplication and is left as an exercise. Conversely, letf be a finite extension field of Q p, and let Φ be a subset of the set Σ F of all embeddings ι: F C p. Let E = E(F, Φ) be the reflex field of the p-adic CM-type (F, Φ), i.e. E = Q p ( ι Φ ι(x)) x F. Equivalently, E is the field of definition of the character ι Φ ι as a character of F, or the field of definition of the cocharacter h Φ : G mcp T F Cp of T F attached to Φ. (5.7) Proposition Let E 1 be a finite extension field of E = E(F, Φ), where (F, Φ) is a p-adic CM-type. Let K = K 0 E 1, the completion of the maximal unramified extension of E 1 in K. (i) There exists a p-divisible group X over the ring of integers O K of K with height(x ) = [F : Q p ], plus a homomorphism β : O F End(X ), such that the character of the action of F on Lie(X ) OK K is ι Φ ι. (ii) Any two p-divisible groups X 1 and X 2 over O K with the same CM-type (F, Φ) are O F - linearly isomorphic. (iii) Given any p-divisible group X over O K satisfying the properties in (i), any element u O F and any uniformizer π E 1 of E 1, there exists a p-divisible group Y over O E1 with an action γ : O F End(X by O F such that (Y, γ) Spec(OE1 ) Spec(O K ) is O F - linearly isomorphic to (X, β) and ρ Y (rec E1 (π E )) = u. Here ρ Y : Γ E1 O F is the Galois representation attached to Y and rec E1 : E 1 Γ E 1 is the arithmetically normalized local Artin map. Proof. The statement (iii) is a consequence of unramified descent. Because O F is a complete discrete valuation ring, it suffices to prove (i) and (ii) up to O F -linear isogeny; let (i) and (ii) be the isogeny statement corresponding to (i) and (ii) respectively. Then (i) and (ii) follow from 5.2 (ii) and 5.3 (iii): the crystalline p-adic Galois representation corresponding to X is the object corresponding to the homomorphism H ab K can T E δ F,Φ T F, where δ F,Φ is the homomorphism between induced tori attached to the CM-type (F, Φ) described in 5.6, and can: H ab K T E is the canonical surjection defined by the Lubin-Tate formal group law for E. 11

12 Remark. In terms of Galois representations, (iii) asserts that every algebraic abelian Galois representation I E1 O F of the decomposition group I E can be extended to a continuous homomorphism from Γ E1 to O F. This is clear because Γ E 1 = IE1 Gal(κ E1 /κ E1 ). (5.8) Corollary Notation as in 5.7. There exists a p-divisible group Y over O E1 with action by O F of CM-type (F, Φ) such ρ Y (Γ E1 ) = ρ Y (I E1 ) as subgroups of O F, where ρ Y : Γ E O F is the Galois representation attached to Y. Proof. Immediate from 5.7 (iii). Let E be a finite extension field of Q p. Let κ E = Fq be the residue field of E, q = p r, and let π E be a uniformizer of E. Denote bye 0 the maximal absolutely unramified subextension in E. Let (V, ρ) Let D = D crys (V ), an E 0 -vector space, with the action of a σ-linear Frobenius φ. Let φ r = φ r, an E 0 -linear endomorphism of D. Let δ : T E GL(V ) be the Q p -rational homomorphism such that δ 1 O E ( rec O E ) 1 : IE GL(V ) is equal to the restriction of ρ to the decomposition group I E. Here rec E : E Γ E is the arithmetically normalized local Artin map. (5.9) Proposition (i) Notation as in the preceding paragraph. Then φ 1 r is the E 0 -linear endomorphism of D corresponding to the endomorphism of (V, ρ) given by ρ(rec E (π E )) δ(π E ). (ii) In the case when (V, ρ) is the p-adic Tate module corresponding to a p-divisible group X over O E, (i) asserts that the geometric q-frobenius endomorphism of the κ E -fiber X of X is induced by the endomorphism of the p-divisible group X corresponding to the endomorphism of (V, ρ) given by the element ρ(rec E (π E )) δ(π E ) of GL(V ). Proof. We give a sketch of a proof below; see [2, 6.3] for details. First we note that the expression ρ(π E ) δ(π E ) 1 is independent of the choice of the uniformizer π E. Observe first that the assertion (i) is compatible with unramified twists, i.e. it suffices to prove the assertion for (V, ρ χ) where χ: Gal(κ E /κ E ) End ρ (V ). Moreover (i) is compatible with tensor products. So it suffices to check the assertion (i) for Lubin-Tate formal group laws, which is a well-known fact recall in 5.4. The statement (ii) follows immediately from (i). We remark that (ii) is a standard fact in the situation of complex multiplication of abelian varieties; see for instance [12, 7]. (5.10) Proposition Let X be a simple p-divisible group over a finite field F q. Let F be a maximal commutative semisimple subalgebra of End 0 (X), and let Φ Σ F be a CM-type of F such that card(φ)/[f : Q p ] is equal to the slope of X. Let E be the reflex field attached to (F, Φ), and let E 1 be a finite extension field of E such that the residue field κ E1 = Fq r for some r N. Then there exists a p-divisible group Y over O E1 and with action by O F such that the closed fiber Y := Y Spec(OE1 ) Spec(κ E1 ) is O F -linearly isogenous to X Spec(Fq) Spec(κ E1 ). Proof. By 5.7 and 5.9, there exists an O F -linear p-divisible group Y over O E1 such that the Frobenius element of its closed fiber is equal to πx r. Hence the closed fiber is O F -linearly isogenous to isogenous to X Spec(Fq) Spec(κ E1 ) by Tate s conjecture for p-divisible groups over finite fields. Remark. Prop is a reformulation of the result in [2, 6]. 12

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