On deformations of $1$-motives

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On deformations of 1-motives

A. Bertapelle, Nicola Mazzari

To cite this version:

A. Bertapelle, Nicola Mazzari. On deformations of

pp.11 - 22. �10.4153/CMB-2017-076-2�. �hal-01814268�

http://dx.doi.org/10.4153/CMB-2017-076-2

© Canadian Mathematical Society 2018

On Deformations of 1 -motives

A. Bertapelle and N. Mazzari

Abstract. According to a well-known theorem of Serre and Tate, the infinitesimal deformation the- ory of an abelian variety in positive characteristic is equivalent to the infinitesimal deformation theory of its Barsotti–Tate group. We extend this result to 1-motives.

1 Introduction

LetRbe an artinian local ring with maximal idealmand perfect residue fieldkof positive characteristicp. LetM¹(R)denote the category of (smooth) 1-motives over R. For any 1-motiveM(resp. Barsotti–Tate groupB) overR, letM₀(resp.B⁰) denote its base change tok = R/m. Let Def(R,k)denote the category whose objects are triples(M₀,B,ε₀)whereM₀is a 1-motive overk,Bis a Barsotti–Tate group overR andε₀∶B⁰→M₀[p^∞]is an isomorphism ofB⁰with the Barsotti–Tate group ofM₀. The aim of this paper is to prove the following theorem.

Theorem 1.1 The functor

∆_R∶M¹(R) Ð→Def(R,k) (1.1)

Mz→ (M₀,M[p^∞], natural ε₀), is an equivalence of categories.

This result generalises the well-known Serre–Tate equivalence for abelian schemes over artinian local ringsRas above (cf.[6, Theorem 1.2.1], [8, V, Theorem 2.3]). Argu- ments in [6,8] do not extend directly to the case of 1-motives but are used to get some intermediate results. Note that the proof of fullness (Proposition4.3) and essential surjectivity (Proposition4.5) uses weights, a careful study of the case of 1-motives of the form[Z→G^m](Lemma2.4), which is orthogonal to the classical case of abelian schemes, and Galois descent arguments (Lemma4.4). Another essential ingredient is that tori and étale group schemes overklift uniquely toR.

As an application, in Section4.3we extend the description of the formal mod- uli space of an ordinary abelian variety over an algebraically closed field and its Serre–Tate coordinates to 1-motives. Consequently, canonical liftings of (ordinary) 1-motives overkto 1-motives overW(k)exist (Proposition4.10).

Received by the editors October 10, 2017; revised November 17, 2017.

Published electronically March 22, 2018.

AMS subject classification: 14L15, 14C15, 14L05.

Keywords: 1-motive, Barsotti-Tate group.

Notation

Letp^s,s≥1, denote the characteristic ofR. ThenRis canonically endowed with the structure of a finiteW_s(k)-algebra, whereW_s(k) = W(k)/(p^s)denotes the ring of Witt vectors of lengths. We also fix a positive integernsuch that(1+m)^pn= {1}. For anyR-group schemeG, we identifyG(R) =Hom_R(SpecR,G)with Hom_R-gr(Z,G) by mappinga∈G(R)to the morphismu∶Z→Gsuch thatu(1) =a.

2 Barsotti–Tate Groups and 1 -motives

LetM¹(R)be the category of (smooth) 1-motives overR. Its objects are two term complexes of commutativeR-group schemesM = [u∶L → G]in degrees−1 and 0, whereG is extension of an abelian schemeAby a torusT, andLis locally for the étale topology onRisomorphic toZ^rfor some non-negative integerr. Recall that a 1-motive has a natural weight filtration

0⊆W₋₂M= [0→T] ⊆W₋₁M= [0→G] ⊆W₀M=M.

Since morphisms of 1-motives respect filtrations,M¹(R)is a filtered category. We will denote byM¹(R)_≤i the full subcategory ofM¹(R)consisting of 1-motives such that M=W_iM. Given a 1-motiveM, letM_ab= [u_ab∶L→A], whereu_abis the composition ofuwith the canonical morphismG →A. LetM¹(R)_≥−1be the full subcategory of M¹(R)consisting of 1-motivesM=M_ab. Similarly Def(R,k)_≥−1(resp. Def(R,k)_≤−1) denotes the full subcategory of Def(R,k)consisting of objects such thatM₀ is in M¹(k)_≥−1(resp. inM¹(k)_≤−1).

For anym∈N, consider the cone of the multiplication bymonM:

M/mM∶L ⁽

−u

−m)

ÐÐ→G⊕L ⁽⁻

m,u)

ÐÐÐÐ→G,

whereLis in degree−2, and letM[m] = H⁻¹(M/mM)(see [4, 10.1.4] or [2, §1.3]

with a different sign convention); it is a finite and flatR-group scheme and it fits in the diagram

(2.1) L

 _(_−m^−u)

L _

−m

̃η_M[m]∶ 0 //G[m] //Ker((−m,u))

//L //

0

η_M[m]∶ 0 //G[m] //M[m] //L/mL //0.

Remark 2.1 Let us consider

ξ_G[m]∶0Ð→G[m] Ð→GÐ→^m GÐ→0.

By direct computations one gets̃η_M[m]= −u^∗ξ_G[m]as elements of Ext_R(L,G[m]). Since the push-out of the extension ̃η_M[pr] along the canonical morphism G[p^r] → G[p^r+1]is isomorphic to p⋅ ̃η_M[pr+1], there is a morphism of complexes

η_M[pr]→η_M[pr+1]for anyr≥1. One then gets a short exact sequence of Barsotti–Tate groups (BT groups for short)

(2.2) η_M[p∞]∶0Ð→G[p^∞] Ð→M[p^∞] Ð→L[p^∞] Ð→0, whereL[p^∞] ∶=L⊗_ZQ^p/Z^p.

Lemma 2.2 Let M₀ = [u₀∶L₀→G₀], let M^′₀be1-motives over k, and let φ₀∶M₀→ M₀^′be a morphism of1-motives. Denote by T₀the maximal subtorus of G₀.

(i) Any finite and flat R-group scheme F that lifts M₀[m]is endowed with a unique weight filtration that lifts the weight filtration T₀[m] ⊆G₀[m] ⊆M₀[m]. (ii) Any morphism of finite flat R-group schemes F → F^′ that lifts φ₀[m]respects

filtrations.

(iii) Any BT group over R that lifts M₀[p^∞]admits a unique filtration that lifts the weight filtration on M₀[p^∞]and any morphism of BT groups over R which lifts φ₀[p^∞]respects filtrations.

(iv) If M₀ lifts to a1-motive M over R, then the natural weight filtration on M[m] agrees with the one induced by the weight filtration on M₀.

Proof LetFbe a finite flatR-group scheme liftingM₀[m]. LetLbe the unique étale lifting ofL₀overR. We claim thatFfits into a short exact sequence

0Ð→EÐ→FÐ→L/mLÐ→0, which lifts the sequenceη_M

0[m]forM₀in diagram (2.1). Indeed, there exists a unique lifting f∶F → L/mLof the morphismM₀[m] →L₀/mL₀, sinceL/mLis étale, and we setE=Kerf. Note further thatf factors through the epimorphismπ_F∶F→F^ét= F/F^○withF^○the identity component ofF. Let f^ét∶F^ét→L/mLsatisfy f^ét○π_F = f. The morphismf₀^étis an epimorphism of finite étalek-group schemes, hence so is f^ét. It follows thatf is an fppf epimorphism, hence the short exact sequence as claimed.

LetL^∗₀andF^∗be the Cartier duals ofL₀andF, respectively, and letTbe the unique torus liftingT₀. We have seen that the canonical morphismM₀[m]^∗→L^∗₀/mL^∗₀lifts uniquely to a morphism F^∗ → L^∗/mL^∗ overRand hence, by Cartier duality, the immersionT₀[m] →M₀[m]lifts uniquely to an immersionT[m] →FoverR. The compositionT[m] →F→L/mLis the zero map, since its reduction modulomis the zero map andL/mLis étale. The closed immersionT[m] →Ffactors thus through E. By construction, the filtration

0⊆T[m] ⊆E⊆F

ofFlifts the weight filtration on M₀[m], so the existence of a filtration as in (i) is proved.

With similar arguments, one shows statement (ii). Statement (iv) and the unique- ness of the filtration in (i) follow from (ii), and statement (iii) is a formal consequence of (i) and (ii).

The classical Serre–Tate theorem states that deformations of an abelian variety over konly depend on deformations of its BT group. As we will now explain, the analo- gous result for 1-motives of the type[Z→G^m,k]can explicitly be deduced from the

canonical isomorphism

(2.3) R^{∗ ∼}Ð→k^∗× (1+m), xz→ (x₀,x/[x₀]),

wherex₀is the reduction ofxmodulomand[x₀] ∈W_s(k)is the multiplicative rep- resentative ofx₀. Recall that(1+m)^pn = {1}, and hence 1+m=µ_pn(R) =µ_p∞(R).

We now consider BT groups that are extensions ofQ^p/Z^pbyµ_p∞. By [8, Proposi- tion 2.5, p. 180], the map

Ψ∶1+mÐ→Ext_R(Q^p/Z^p,µ_p∞),

which associates withu∈1+mthe push-out alongu∶Z→µ_p∞of the sequence 0Ð→ZÐ→Z[p⁻¹] Ð→Q^p/Z^pÐ→0

is an isomorphism. In particular, pⁿkills Ext_R(Q^p/Z^p,µ_p∞). For anyu ∈ R^∗, let M^u= [u∶Z→G^m,R]and consider the homomorphism

(2.4) Φ∶1+mÐ→Ext_R(Q^p/Z^p,µ_p∞), which mapsu∈1+m⊂R^∗to the extensionη_Mu[p∞]as in (2.2).

Lemma 2.3 We haveΦ= −Ψ; hence,Φis an isomorphism.

Proof Let Ext_Z/pn

Z( ⋅,⋅ ) denote classes of extensions of pⁿ-torsion R-group schemes. By [8, p. 183], we have isomorphisms

Ext_R(Q^p/Z^p,µ_p∞) ≃lim

←Ð

m

Ext_R(Z/p^mZ,µ_p∞) (2.5)

≃Ext_R(Z/pⁿZ,µ_p∞) ≃Ext_Z/pn

Z(Z/pⁿZ,µ_pn),

where the first isomorphism is induced by pull-back alongZ/p^mZ → Q^p/Z^p, the second isomorphism follows, since pⁿ kills Ext_R(Q^p/Z^p,µ_p∞), and the third iso- morphism is obtained by passing to kernels of the multiplication bypⁿ, sinceµ_p∞is divisible. Note that the composition of the isomorphisms in (2.5) associates with an extension ofQ^p/Z^pbyµ_p∞the corresponding sequence of kernels of the multiplica- tion bypⁿ. Let

Ψ_n, Φ_n∶1+mÐ→Ext_Z/pn

Z(Z/pⁿZ,µ_pn)

denote the composition of Ψ (resp. Φ) with the isomorphisms in (2.5). We are left to prove that Φ_n= −Ψ_n.

Note that Ψ_n(u)can also be constructed taking first the push-out alongu∶Z → G^m,Rofζ∶0→Z→Z→Z/pⁿZ→0 and then considering the sequence of kernels of the multiplication by pⁿ. Indeed,u_∗ζ = ι^∗_nι_∗Ψ(u)withι_n∶Z/pⁿZ→Q^p/Z^pand ι∶µ_p∞ →G^m,Rthe usual morphisms. Since Cartier duality induces the identity both on 1+m=Hom_R-gr(Z,G^m)and on Ext_Z/pn

Z(Z/pⁿZ,µ_pn), Ψ_n(u)is also represented by the sequence of cokernels of the multiplication bypⁿof the sequence obtained via pull-back alongu∶Z→G^m,Rfrom the sequenceξ_µ

pn∶0→µ_pn →G^m,R →G^m,R→0.

On the other hand, Φ_n(u)is the sequenceη_Mu[pn]in diagram (2.1). By Remark2.1 and diagram (2.1), we conclude that Φ_n(u) = −Ψ_n(u).

We now see how to interpret (2.3) in terms of deformations of 1-motives.

Lemma 2.4 Given a1-motive M₀= [u₀∶Z→G^m,k]and aB∈Ext_R(Q^p/Z^p,µ_p∞) there is a unique1-motive M = [u∶Z→ G^m,R]that lifts M₀and whose BT group is isomorphic toBas extension of µ_p∞byQ^p/Z^p.

Proof Note that ifB,B^′are two liftings ofM₀[p^∞], there is at most one morphism B→B^′as extensions ofQ^p/Z^pbyµ_p∞. We have to prove that the homomorphism

R^∗Ð→k^∗×Ext_R(Q^p/Z^p,µ_p∞), uz→ (u₀,η_Mu[p∞])

is an isomorphism. Note that ifu= [u₀],u₀ ∈k^∗, thenuadmitsp^r-th roots for any r ≥1, sincekis perfect; hence,η_Mu[p∞]is split, since Ext_R(Q^p/Z^p,µ_p∞)is killed by pⁿ. We are then reduced to checking that the homomorphism

1+mÐ→Ext_R(Q^p/Z^p,µ_p∞), uz→η_Mu[p∞],

is an isomorphism. But this homomorphism equals Φ in (2.4), and the latter is an isomorphism by Lemma2.3.

3 Auxiliary Results

3.1 Galois Actions

Letk^′/kbe a finite Galois extension of Γ=Gal(k^′/k)and setR^′=R⊗W(k)W(k^′). Note thatR^′is an artinian local ring with residue fieldk^′and SpecR^′→ SpecRis a Galois covering of group Γ. Then Γ naturally acts onM¹(R^′)and Def(R^′,k^′)and the Serre–Tate functor ∆_R′(1.1) commutes with the Galois action.

Note that the datum of a 1-motiveMoverRis equivalent to the datum of a 1-motive M^′= [u^′∶L^′→G^′]overR^′with a Γ-action compatible with the Γ-action on SpecR^′. Indeed,L^′descends to a lattice overR. Further, since the topological space underlying G^′coincides with the topological space underlying the semi-abeliank^′-varietyG₀^′, it can be covered by affine open subschemes which are stable under the Γ-action and hence it descends to anR-group scheme. The maximal subtorusT^′ofG^′descends to a subtorusT ofG andG/T is an abelian scheme, since it is an abelian scheme after base change toR^′. Similarly, the datum of an object(G₀,B,ε₀)in Def(R,k) is equivalent to the datum of a(G^′₀,B^′,ε₀^′)in Def(R^′,k^′)together with a Γ-action compatible with the Γ-action onk^′(in the first and third entries) and the Γ-action on R^′(on the second entry).

3.2 Refinements

Recall that any 1-motiveM= [u∶L→G]overRadmits a so-called universal extension M^♮ = [u^♮∶L → G^♮]that fits into a short exact sequence of complexes of R-group schemes

(3.1) 0Ð→ [0Ð→V(M)] Ð→M^♮Ð→MÐ→0,

whereV(M)is the vector group associated to the dual sheaf of Ext¹_Zar(M,G^a,R). Note thatV(M)is killed byp^s, since the multiplication byp^smorphism is the 0 map on G^a,R. Note thatM^♮is denoted byE(M) = [L→E(M)G]in [1,2].

Remark 3.1 We can refine some preliminary results in [6]. Lets^′≥sbe the minimal integer such thatm^s′=0.

(i) LetGbe any smooth commutativeR-group scheme. By [6, Lemma 1.1.2] the kernel of the reduction mapG(R) → G(k)is killed byp^s(s′−1). In our hypothesis one can prove that it is killed byp^s′−1. Indeed, by the theory of Greenberg functor the sectionsG(R/mⁱ), 1≤i≤s^′, can be identified with the group ofk-rational sections of a smooth k-group scheme Gr_i(G). Further, there are so-called change of level morphismsρ¹_i∶Gr_i+1(G) →Gr_i(G)such thatρ¹_i(k)coincides with the reduction map G(R/mⁱ⁺¹) →G(R/mⁱ). By Greenberg’s structure theorem ([5],[3, Thm. A.17]) the kernel ofρ¹_i is ak-vector group, thus killed by p. It follows then by induction that Ker(G(R) →G(k))is killed byp^s′−1.

(ii) By [6, Lemma 1.1.3 (3)] a morphism of semi-abelian varietiesφ₀∶G₀→H₀lifts overRup to multiplication byp^s(s′−1). We can also improve this estimate ifp>2. Let M,Nbe 1-motives overR. By [1, Thm. 2.1] there exists a canonical morphism between universal extensions (3.1)φ^♮∶M^♮→ N^♮that liftsφ^♮₀. Ifφ^♮(V(M)) ⊆V(N), thenφ^♮ induces a morphismφ∶M → Nthat liftsφ₀. In general, since the multiplication by p^smorphism killsV(M), the morphismp^sφ^♮mapsV(M)to 0. Hence,p^sφ₀lifts to a morphism “p^sφ”∶M→N.

4 Proof of the Main Theorem

4.1 Full Faithfulness

Proposition 4.1 Let M,N be two1-motives over R. Then the reduction map Hom_M

1(R)(M,N) Ð→Hom_M

1(k)(M₀,N₀) is injective.

Proof Consider a morphismφ∶M→Nand assume that its reduction modulomis the 0 morphism. Ifp>2, thenφ=0, since the induced morphism between universal extensionsφ^♮∶M^♮ → N^♮is the zero map [1, Thm. 2.1]. In general, one hasφ=0 in degree−1 by the equivalence of the étale sites overRand overk, andφ=0 in degree 0 by [6, Lemma 1.1.3 2)].

Corollary 4.2 The functor(1.1)is faithful.

Proposition 4.3 The functor(1.1)is full.

Proof LetM= [u∶L→G],N= [v∶F→H]be two 1-motives overR,φ₀∶M₀ →N₀ a morphism between their reduction modulom, andψ∶M[p^∞] →N[p^∞]a lifting of φ₀[p^∞]∶M₀[p^∞] →N₀[p^∞]. We have to prove that there exists a liftingφ∶M →N ofφ₀overRsuch thatφ[p^∞] =ψ.

As a first step, consider the caseL=F=0. We assume thatp>2; ifp=2, the same proof works replacingswiths(s^′−1), wherem^s′=0, and it coincides with the one in [6, p. 144]. Given aφ₀∶G₀→H₀and a morphismψ∶G[p^∞] →H[p^∞]liftingφ₀[p^∞], by Remark3.1(ii) there exists a morphism “p^sφ”∶G→Hliftingp^sφ₀. Further,p^sψ=

“p^sφ”[p^∞]by [6, Lemma 1.1.3(2)], since both morphisms liftp^sφ₀[p^∞]. Hence, “p^sφ”

killsG[p^s]and thus “p^sφ” = p^sφfor a morphismφ∶G → H (necessarily unique since Hom_R-gr(G,H)has nop-torsion). Thus, the restriction of the functor (1.1) to M¹(R)_≤−1is full.

As a second step, consider the case whereGandHare abelian varieties. Fullness of the restriction of the functor (1.1) toM¹(R)_≥−1follows from the previous step via Cartier duality.

For the general case, letM = [u∶L→G],N= [v∶F →H]be two 1-motives over R,φ₀∶M₀ → N₀ a morphism between their reduction modulomandψ∶M[p^∞] → N[p^∞]a lifting ofφ₀[p^∞]∶M₀[p^∞] → N₀[p^∞]. Let φ₀ = (f₀,g₀), i.e., f₀∶L₀ → F₀,g₀∶G₀ → H₀and g₀○u₀ =v₀○ f₀. Recall that any liftingψofφ₀[p^∞]respects weight filtrations by Lemma 2.2. Hence, by the first step of the proof there exists a unique morphismg∶G → Hliftingg₀. Further, by the equivalence between the category of étale group schemes overkand the category of étale group schemes over R, there exists a uniquef∶L→Fliftingf₀. We are left to prove thatg○u=v○f, so thatφ= (f,g)is a morphism of 1-motives, and thatφ[p^∞] =ψ. The latter equality follows from [6, Lemma 1.1.3 2)], since both morphisms are liftings ofφ₀[p^∞].

LetZ= [w∶L→H]withw=g○u−v○f. We claim thatη_Z[p∞]in (2.2) is split.

For proving this claim, it is sufficient to check thatη_Z[pr](and hencẽη_Z[pr]in (2.1)) is split for anyr. Sinceψ[p^r]restricts to the morphism g[p^r]∶G[p^r] → H[p^r]on weight≤ −1 subgroups and induces f/p^rf∶L/p^rL→F/p^rFin weight 0, it is

(f/p^rf)^∗η_N[pr]=g[p^r]_∗η_M[pr], and hence

f^∗̃η_N[pr]=g[p^r]_∗̃η_M[pr]. By Remark2.1, we have

̃η_Z[pr]= (−w)^∗ξ_H[pr]=f^∗(v^∗ξ_H[pr]) −u^∗(g^∗ξ_H[pr]) = f^∗̃η_N[pr]−u^∗g[p^r]_∗ξ_G[pr]

=f^∗̃η_N[pr]−g[p^r]_∗̃η_M[pr]=0.

Hence,Zis a 1-motive such thatη_Z[p∞]is split extension ofL[p^∞]byH[p^∞]and w₀ = 0. We are left to prove thatw = 0. We can work étale locally onRand then assume thatLis constant and the maximal subtorusSofHis split. By the second step of this proof,w_ab∶L → Bis the 0 morphism. Hence,wfactors throughS. Let M_t= [w∶L→S]and note thatM_t[p^∞]is split extension ofL[p^∞]byS[p^∞]. Hence, w=0 by Lemma2.4.

4.2 Essential Surjectivity

The strategy of the proof of essential surjectivity is first to construct the desired 1-mo- tive étale locally and then to apply descent. We then need the following result.

Lemma 4.4 Let k^′/k be a finite Galois extension of Galois groupΓand set R^′ = R ⊗W(k)W(k^′). A1-motive M^′ over R^′descends to R if and only if its image in Def(R^′,k^′)via(1.1)descends toDef(R,k).

Proof The necessity is clear. For sufficiency, letM^′= [u^′∶L^′→G^′]and assume that (M^′₀,B^′=M^′[p^∞], can∶B^′0

∼

→M₀^′[p^∞])descends to an object (M₀= [L₀→G₀],B,ε₀∶B⁰Ð→^∼ M₀[p^∞]))

in Def(R,k). For anyσ ∈Γ we then have an isomorphism in Def(R^′,k^′) (φ_σ,0,ψ_σ)∶ (M^′₀,B^′, can)Ð→ (^∼ σ^∗M₀^′,σ^∗B^′,σ^∗can), where

φ_σ,0∶M^′₀Ð→^∼ σ^∗M₀^′, ψ_σ∶B^{′ ∼}Ð→σ^∗B^′, make the diagram

B^′0

ψ_σ,0

//

can

σ^∗B^′0 σ^∗can

M₀^′[p^∞]

φ_σ,0[p^∞] //σ^∗M^′₀[p^∞]

commute. By the full faithfulness of (1.1) proved in Corollary4.2and Proposition4.3, the isomorphism(φ_σ,0,ψ_σ)gives a unique isomorphismφ_σ∶M^{′ ∼}→ σ^∗M^′that lifts φ_σ,0and restricts toψ_σ on BT groups. Hence, we have defined an action of Γ onM^′ compatible with the Γ-action on R^′and thusM^′descends overR, as explained in Section3.1.

Proposition 4.5 The functor(1.1)is essentially surjective.

Proof Let(M₀= [L₀→G₀],B,ε₀)be an object of Def(R,k).

As a first step, consider the case whereL₀=0. Thanks to Lemma4.4we can assume that the maximal subtorusT₀ofG₀is split of dimensiond. The Cartier dual ofG₀is a 1-motive[w₀∶Z^d→A^∗₀], whereA^∗₀is the dual of the abelian quotientA₀ofG₀. The abelian varietyA₀lifts to an abelian schemeAoverR, and since the reduction map A(R) →A(k)is surjective, the morphismw₀lifts to a morphismw∶Z^d→A^∗. Passing to Cartier duals, we obtain anR-group schemeGthat is an extension ofAbyG^dm,R

and liftsG₀. Now, the BT groupG[p^∞]might not be isomorphic toB. Repeating word by word the proof of the classical Serre-Tate theorem [6, Thm. 1.2.1, pp. 145–

146], one finds a finite flat subgroup schemeKofG[p^2s]such thatG/Kis a lifting of G₀with BT group isomorphic toB.

The case whenM₀=M_0,abfollows immediately by Cartier duality from the previ- ous step.

For the general case, we can assume thatL₀is constant and the maximal torus in G₀is split, again by Lemma4.4. Recall that by Lemma2.2the BT groupBis naturally filtered so thatW₋₁Bis a lifting ofG₀[p^∞]andB/W₋₂Bis a lifting ofM_0,ab[p^∞]. By the previous steps we know thatG₀lifts to anR-schemeG, which is an extension of an abelian schemeAbyG^dm,R, andG[p^∞] =W₋₁B; furtherM_0,ablifts to a 1-motive M_A= [u_A∶Z^m→A]whose BT group is isomorphic toB/W₋₂B.

LetM^′= [u^′∶Z^m →G]be any extension ofM_AbyT; it exists, sinceH¹(R,G^m,R) = 0. SinceT(R) →T(k)is surjective, we can assume thatu^′is also a lifting ofu₀. We are then left to alteru^′so thatM^′[p^∞] ≃B[p^∞]in Ext_R((Q^p/Z^p)^m,G[p^∞]).

LetE=B[p^∞] −M^′[p^∞], as extension of(Q^p/Z^p)^mbyG[p^∞]. Since the push- out alongG[p^∞] →A[p^∞]mapsEto the trivial extension of(Q^p/Z^p)^mbyA[p^∞], there exists by Lemma2.4a 1-motiveN = [v∶Z^m → T]such thatv₀ = 0 and the push-out ofN[p^∞]alongT[p^∞] →G[p^∞]isE. ThenM= [u=u^′+v∶Z^m →G]is a lifting ofM₀andM[p^∞]is isomorphic toB[p^∞].

With this proposition, the proof of Theorem1.1is completed.

Remark 4.6 Note that some results on deformations of 1-motives (but not Theo- rem1.1) were proved with other methods in Madapusi’s thesis [7]; however, they are not included in the preprint written by the author under the name K. Madapusi Pera and bearing the same title of the thesis.

Since any extension of an étale BT group by a toroidal BT group is split overk, we deduce from Theorem1.1the following generalization of Lemma2.4.

Corollary 4.7 Suppose T is an R-torus and L is a lattice. Given a 1-motive M₀= [u₀∶L₀→T₀]and any BT groupBthat is an extension of L[p^∞]by T[p^∞]there is a unique1-motive M= [u∶L→T]that lifts M₀and whose BT group is isomorphic toB.

4.3 Formal Moduli and Serre–Tate Coordinates

Letkbe an algebraically closed field of characteristicp >0. We say that a 1-motive M₀overkis ordinary if gr

−1M₀=A₀is an ordinary abelian variety (possibly trivial).

Thanks to Theorem1.1one can easily extend Serre–Tate theory for ordinary abelian varieties to ordinary 1-motives. In this section, we will illustrate some results in this direction. Proofs are only sketched, since no new phenomena appear.

Following [6, § 2] one can define the formal moduli of a given 1-motiveM₀over k. Namely, letM̂^M₀=M̂be the functor

M̂(R) ∶= {R-liftings ofM₀}/iso,

whereRis an artinian local ring with residue fieldk. By Theorem1.1we get a bijection M̂(R) = {R-liftings ofM₀[p^∞]}/iso.

Theorem 4.8 Let M₀, N₀be ordinary1-motives over k.

(i) Let R be an artinian local ring as above. For any1-motive M over R lifting M₀, there exists a canonicalZ^p-bilinear form (the Serre–Tate coordinates)

q(M/R,⋅,⋅ )∶T_pM₀(k) ⊗T_pM^∗₀(k) Ð→ ̂G^m(R), where M₀^∗denotes the Cartier dual of M₀and T_pM₀(k) ∶=lim

←Ðⁿ

M₀[pⁿ](k). It induces an isomorphism of functors

M̂( ⋅ ) ≃Hom

Zp(T_pM₀(k) ⊗T_pM^∗₀(k),̂ G^m( ⋅ )).

(ii) Let φ₀∶M₀ →N₀ be a morphism of1-motives and let M, N be liftings over R of M₀, N₀, respectively. Then φ₀lifts to an R-morphism φ∶M→N if and only if

q(M/R,α,φ^∗₀(β)) =q(N/R,φ₀(α),β)

for every α∈T_pM₀(k)and β∈T_pM^∗₀(k). Further, if a lifting exists, it is unique.

Proof The proof goes exactly as in the classical case. For the convenience of the reader we point out the main steps (cf.[6, p. 152]).

● Sincekis algebraically closed andM₀is ordinary, the BT groupsM₀[p^∞]^○is split multiplicative andM₀[p^∞]^étis split étale. Hence, the same are true forM[p^∞]^○and M[p^∞]^étfor any liftingMofM₀ overR. For this reason we may writeM₀[p^∞]^○ (resp.M₀[p^∞]^ét) for the corresponding functor on the category of artinian local rings with residue fieldk.

● By Cartier duality, there is a perfect pairingM₀[pⁿ] ×M^∗₀[pⁿ] → µ_pn [4, 10.2.5], inducing an isomorphism of functors

M₀[p^∞]^{○ ∼}Ð→Hom

Zp(T_pM^∗₀(k),Ĝ^m),

on the category of artinian local rings with residue fieldk. We denote the corre- sponding pairing byE_M∶M₀[p^∞]^○×T_pM₀^∗(k) → ̂G^m.

● By [8, Proposition 2.5 p. 180] and the previous steps, for anyRas above and any liftingMofM₀overR, we have isomorphisms

Ext_R(M[p^∞]^ét,M[p^∞]^○) =Ext_R(T_pM₀(k) ⊗Q^p/Z^p,M[p^∞]^○)

≃Hom(T_pM₀(k),M[p^∞]^○(R));

thus, there exists a uniqueϕ_M ∈ Hom_R(T_pM₀(k),M[p^∞]^○(R))associated with the class of the BT groupM[p^∞], viewed as extension ofM[p^∞]^étbyM[p^∞]^○. Then we can defineq(M/R,α,β) ∶= E_M(ϕ_M(α),β)in̂

G^m(R)and follow word by word the proof in [6].

Remark 4.9 LetM₀be an ordinary 1-motive overKand denote its graded quo- tients (by the weight filtration) byL₀,A₀ andT₀. Fix basis: α₁, . . . ,α_gofT_pA₀(k) withg=dimA₀;α_g+1, . . . ,α_g+ℓofT_pL₀(k)withℓ=rankL₀;β₁, . . . ,β_gofT_pA^∗₀(k); β_g+1, . . . ,β_g+t ofT_pT₀^∗(k)witht =dimT₀. Then the coordinate ring ofM̂is iso- morphic to

W(k)[[T_i,j]] whereT_i,j=q(α_i,β_j) −1,

whereq( ⋅.⋅ )is the bilinear form associated with the universal formal deformation.

In particularM̂is represented by a(g+ℓ) × (g+t)-dimensional formal torus.

Proposition 4.10 Let M₀be an ordinary1-motive over k. Then there exists a1-motive M over W(k)lifting M₀such thatEnd_M

1(W(k))(M) →End_M

1(k)(M₀)is bijective.

Proof By Theorem1.1there exists a unique 1-motiveM_n= [L_n→G_n]overW_n+1(k) liftingM₀such thatM_n[p^∞]has split connected-étale sequence. Asngoes to infinity, these liftings form a compatible system. In order to see that the limit is algebraizable it

is enough to check that lim Ð→ⁿ

G_nis algebraizable, since allL_nare torsion-free constant abelian groups of the same rank, and for any group schemeJof finite type overW(k)

(4.1) J(W(k)) =lim

←Ð

m

J(W_m(k)).

Now, by Cartier duality, the system(G_n)ncorresponds to a compatible system of 1- motives[T_n^∗→A^∗_n]. The latter is algebraizable if the system(A^∗_n)nis. SinceM_n[p^∞] has split connected-étale sequence, the same are true forA_n[p^∞]and A^∗_n[p^∞]. So (A^∗_n)nis algebraizable by the proof of [8, Ch. 5, Thm 3.3, p. 173].

Let us denote byM= [u∶L→G](resp.A) the lift ofM₀(respA₀) overW(k)con- structed in the previous paragraph; thenGis extension ofAby the unique torusTlift- ingT₀. By construction and Theorem1.1, any reduction map End_M

1(W_n+1(k))(M_n) → End_M

1(k)(M₀)is bijective. We are left to check that the map End_M

1(W(k))(M) → lim

←Ð End_M

1(W_n+1(k))(M_n) = End_M

1(k)(M₀)is bijective. Letφ= (f,g)be an endo- morphism ofM,i.e.,we have morphismsf∶L→L,g∶G→Gsuch thatu○f =g○u.

Ifφ₀=0, thenφ=0 by devissage, since

(4.2) End_W(k)(L) =End_k(L₀), End_W(k)(T) =End_k(T₀),

and End_W(k)(A) = End_k(A₀)by [8, Ch. 5, Thm 3.3]. For surjectivity, let φ_n = (f_n,g_n)∶M_n→ M_nbe a compatible system of endomorphisms. By (4.2) and (4.1) it suffices to show the existence ofg∶G→Glifting the morphisms(g_n)n. We can work with the Cartier dual[T^∗→ A^∗]ofGinstead, and lift the Cartier duals of(g_n)n; as above, we can reduce to the caseT^∗=0 and then conclude by [8, Ch. 5, Thm 3.3].

Acknowledgements This work was done while the second author was visiting the University of Padua supported by a “délégation CNRS”. The first author thanks the project PRIN 2015 “Number Theory and Arithmetic Geometry” for financial support.

Both authors thank the referee, whose detailed comments and helpful suggestions improved the exposition of this paper.

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Lecture Notes in Mathematics, 264, Springer-Verlag, Berlin-New York, 1972.

Dipartimento di Matematica “Tullio Levi-Civita”, Università degli Studi di Padova, via Trieste, 63, I-35121 Padova, Italy

e-mail: [email protected]

Institut de Mathématiques de Bordeaux, University of Bordeaux, F-33405 Talence cedex, France e-mail: [email protected]