On deformations of $1$-motives

TL;DR

This paper extends the classical Serre-Tate deformation equivalence from abelian varieties to 1-motives, broadening the understanding of deformation theory in algebraic geometry.

Contribution

It generalizes the Serre-Tate theorem to include 1-motives, providing new insights into their deformation theory in positive characteristic.

Findings

Deformation theory of 1-motives parallels that of Barsotti-Tate groups.

Extension of Serre-Tate theorem to 1-motives.

Enhanced understanding of infinitesimal deformations in algebraic geometry.

Abstract

According to a well-known theorem of Serre and Tate, the infinitesimal deformation theory 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.

Equations

$${\Delta }_R:{\mathcal{M}}_1(R)$$

$$M$$

$$0\subseteq W_{-2}M=[0\to T]\subseteq W_{-1}M=[0\to G]\subseteq W_{0}M=M.$$

$$0\to [0\to \mathbb{V}(M)]\to M^{♮}\to M\to 0$$

$$M/mM:L\stackrel{\left(\genfrac{}{}{0ex}{}{-u}{-m}\right)}{⟶}G\oplus L\stackrel{(-m,u)}{⟶}G,$$

$$L\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\xi }_{G[m]}:0\to G[m]\to G\stackrel{m}{\to }G\to 0$$

$${\eta }_{M[p^{\infty }]}:0\to G[p^{\infty }]\to M[p^{\infty }]\to L[p^{\infty }]\to 0,$$

$$0\to G[m]\to M[m]\to L/mL\to 0$$

$$R^{\ast }\stackrel{\sim }{⟶}{k}^{\ast }\times (1+\mathfrak{m}),x\mapsto (x_0,x/[x_0]),$$

$$1+\mathfrak{m}={\mu }_{p^n}(R)={\mu }_{p^{\infty }}(R).$$

$$\Psi :1+\mathfrak{m}\stackrel{\sim }{\to }{\mathrm{Ext}}_R({\mathbb{Q}}_p/{\mathbb{Z}}_p,{\mu }_{p^{\infty }})$$

$$0\to \mathbb{Z}\to \underset{m}{\underrightarrow{\lim }}\frac{1}{p^m}\mathbb{Z}\to {\mathbb{Q}}_p/{\mathbb{Z}}_p\to 0.$$

$$\begin{gathered} {\mathrm{Ext}}_R({\mathbb{Q}}_p/{\mathbb{Z}}_p,{\mu }_{p^{\infty }})\simeq \underset{m}{\underleftarrow{\lim }}{\mathrm{Ext}}_R(\mathbb{Z}/p^m\mathbb{Z},{\mu }_{p^{\infty }})\simeq \\ {\mathrm{Ext}}_R(\mathbb{Z}/p^n\mathbb{Z},{\mu }_{p^{\infty }})\simeq {\mathrm{Ext}}_{\mathbb{Z}/p^n\mathbb{Z}}(\mathbb{Z}/p^n\mathbb{Z},{\mu }_{p^n}), \end{gathered}$$

$$\Phi :R^{\ast }\to k^{\ast }\times {\mathrm{Ext}}_R({\mathbb{Q}}_p/{\mathbb{Z}}_p,{\mu }_{p^{\infty }}),$$

$${\Phi }_{|1+\mathfrak{m}}:1+\mathfrak{m}\to {\mathrm{Ext}}_R({\mathbb{Q}}_p/{\mathbb{Z}}_p,{\mu }_{p^{\infty }}),$$

$${\Phi }_n:1+\mathfrak{m}\to {\mathrm{Ext}}_{\mathbb{Z}/p^n\mathbb{Z}}(\mathbb{Z}/p^n\mathbb{Z},{\mu }_{p^n})$$

$${\mathcal{M}}_1(R{)}_{\le -1}$$

$$G$$

$$({\phi }_{\sigma ,0},{\psi }_{\sigma }):\left(G_0^{\prime },{\mathcal{B}}^{\prime },\mathrm{can}\right)\stackrel{\sim }{\to }\left({\sigma }^{\ast }{G}_0^{\prime },{\sigma }^{\ast }{\mathcal{B}}^{\prime },{\sigma }^{\ast }\mathrm{can}\right),$$

$${\phi }_{\sigma ,0}:G_0^{\prime }\stackrel{\sim }{\to }{\sigma }^{\ast }{G}_0^{\prime },{\psi }_{\sigma }:{\mathcal{B}}^{\prime }\stackrel{\sim }{\to }{\sigma }^{\ast }{\mathcal{B}}^{\prime },{\phi }_{\sigma ,0}[p^{\infty }]\circ \mathrm{can}=({\sigma }^{\ast }\mathrm{can})\circ {\psi }_{\sigma ,0}.$$

$${\mathrm{Hom}}_{{\mathcal{M}}_1(R)}(M,N)\to {\mathrm{Hom}}_{{\mathcal{M}}_1(k)}(M_0,N_0)$$

$$(f/p^{r}f{)}^{\ast }{\eta }_{N[p^r]}=g[p^r{]}_{\ast }{\eta }_{M[p^r]},$$

$$f^{\ast }{\tilde{\eta }}_{N[p^r]}=g[p^r{]}_{\ast }{\tilde{\eta }}_{M[p^r]}.$$

$${\tilde{\eta }}_{Z[p^r]}=(-w{)}^{\ast }{\xi }_{H[p^r]}$$

$$({\phi }_{\sigma ,0},{\psi }_{\sigma }):\left(M_0^{\prime },{\mathcal{B}}^{\prime },\mathrm{can}\right)\stackrel{\sim }{\to }\left({\sigma }^{\ast }{M}_0^{\prime },{\sigma }^{\ast }{\mathcal{B}}^{\prime },{\sigma }^{\ast }\mathrm{can}\right),$$

$${\phi }_{\sigma ,0}:M_0^{\prime }\stackrel{\sim }{\to }{\sigma }^{\ast }{M}_0^{\prime },{\psi }_{\sigma }:{\mathcal{B}}^{\prime }\stackrel{\sim }{\to }{\sigma }^{\ast }{\mathcal{B}}^{\prime },$$

$${\mathcal{B}}_0^{\prime }\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\hat{\mathcal{M}}(R):=\{R\text{-liftings of }{M}_0\}/\text{iso}$$

$$\hat{\mathcal{M}}(R)=\{R\text{-liftings of }{M}_0[p^{\infty }]\}/\text{iso}.$$

$$\hat{\mathcal{M}}(-)\simeq {\mathrm{Hom}}_{{\mathbb{Z}}_p}(T_p{A}_0(k)\otimes T_p{A}_0^{\ast }(k),{\hat{\mathbb{G}}}_{\mathrm{m}}(-))$$

Full text

1. Introduction

Let $R$ be an artinian local ring with maximal ideal $\mathfrak{m}$ and perfect residue field $k$ of positive characteristic $p$. Let $\mathcal{M}_{1}(R)$ denote the category of $1$-motives over $R$. For any $1$-motive $M$ (respectively Barsotti -Tate group $\mathcal{B}$) over $R$ let $M_{0}$ (respectively $\mathcal{B}_{0}$) denote its base change to $k=R/\mathfrak{m}$. Let $\mathrm{Def}(R,k)$ denote the category whose objects are triples $(M_{0},\mathcal{B},\varepsilon_{0})$ where $M_{0}$ is a $1$-motive over $k$, $\mathcal{B}$ is a Barsotti -Tate group over $R$ and $\varepsilon_{0}\colon\mathcal{B}_{0}\to M_{0}[p^{\infty}]$ is an isomorphism of $\mathcal{B}_{0}$ with the BT group of $M_{0}$. Aim of this paper is to prove the following

1.0.1 Theorem**.**

The functor

[TABLE]

is an equivalence of categories.

This result generalizes the well-known Serre -Tate equivalence for abelian schemes over artinian local rings $R$ as above (cf. [6, Theorem 1.2.1], [8, V, Theorem 2.3]). Arguments in loc. cit. do not extend directly to the case of $1$-motives (neither to the case of semi-abelian varieties, see Theorem 3.2.1) but are used to get some intermediate results. Note that the proof of fullness (Proposition 4.2.2) and essential surjectiveness (Proposition 4.3.3) uses weights, a carefully study the case of $1$-motives of the form $[\mathbb{Z}\to\mathbb{G}_{m}]$ (Lemma 2.0.5), which is orthogonal to the classical case of abelian schemes, and Galois descent arguments (Lemma 4.3.2).

As a consequence, we can extend in Section 4.4 the description of the formal moduli space of an abelian variety with ordinary reduction over an algebraically closed field to $1$-motives.

Notation

Let $p^{s},s\geq 1$, denote the characteristic of $R$. Then $R$ is canonically endowed with a structure of finite $W_{s}(k)$-algebra, where $W_{s}(k)=W(k)/(p^{s})$ denotes the ring of Witt vectors of length $s$. Let $s^{\prime}\geq s$ be the minimal integer such that $\mathfrak{m}^{s^{\prime}}=0$. We also fix a positive integer $n$ such that $(1+\mathfrak{m})^{p^{n}}=\{1\}$. For any $R$-group scheme $G$ we will identify $G(R)=\mathrm{Hom}_{R}(\mathrm{Spec}\,R,G)$ with $\mathrm{Hom}_{R\text{-gr}}(\mathbb{Z},G)$ by mapping $a\in G(R)$ to the morphism $u\colon\mathbb{Z}\to G$ such that $u(1)=a$.

2. Barsotti -Tate groups and $1$-motives

Let $\mathcal{M}_{1}(R)$ be the category of $1$-motives over $R$. Its objects are two term complexes of commutative $R$-group schemes $M=[u\colon L\to G]$ where $G$ is extension of an abelian scheme $A$ by a torus $T$, and $L$ is locally for the étale topology on $R$ isomorphic to $\mathbb{Z}^{r}$. Recall that a $1$-motive is naturally filtered as follows

[TABLE]

Since morphisms of $1$-motives respect filtrations, $\mathcal{M}_{1}(R)$ is a filtered category. We will denote by $\mathcal{M}_{1}(R)_{\leq i}$ the full subcategory of $\mathcal{M}_{1}(R)$ consisting of $1$-motives such that $M=W_{i}M$. Given a $1$-motive $M$, let $M_{\mathrm{ab}}=[u_{\mathrm{ab}}\colon L\to A]$, where $u_{\mathrm{ab}}$ is the composition of $u$ with the canonical morphism $G\to A$. Let $\mathcal{M}_{1}(R)_{\geq-1}$ be the full subcategory of $\mathcal{M}_{1}(R)$ consisting of $1$-motives $M=M_{\mathrm{ab}}$. Similarly $\mathrm{Def}(R,k)_{\geq-1}$ (respectively $\mathrm{Def}(R,k)_{\leq-1}$) denotes the full subcategory of $\mathrm{Def}(R,k)$ consisting of objects such that $M_{0}$ is in $\mathcal{M}_{1}(k)_{\geq-1}$ (respectively, in $\mathcal{M}_{1}(k)_{\leq-1}$).

Any $1$-motive $M=[u\colon L\to G]$ over $R$ admits a so-called universal extension $M^{\natural}=[u^{\natural}\colon L\to G^{\natural}]$ which fits into a short exact sequence of complexes of $R$-group schemes

[TABLE]

where $\mathbb{V}(M)$ is the vector group associated to the dual sheaf of $\underline{\mathrm{Ext}}^{1}_{\mathrm{Zar}}(M,\mathbb{G}_{a,R})$. Note that $\mathbb{V}(M)$ is killed by $p^{s}$ since the multiplication by $p^{s}$ morphism is the [math] map on $\mathbb{G}_{a,R}$. We recall that $M^{\natural}$ is denoted by $\mathbb{E}(M)=[L\to\mathbb{E}(M)_{G}]$ in [1, 2].

For any $m\in\mathbb{N}$ consider the cone of the multiplication by $m$ on $M$

[TABLE]

where $L$ is in degree $-2$, and let $M[m]=H^{-1}(M/mM)$ (see [4, 10.1.4] or [2, §1.3] with a different sign convention); it is a finite and flat $R$-group scheme and it fits in the following diagram

[TABLE]

2.0.1 Remark**.**

One checks by direct computations that the pull-back of

[TABLE]

along $u$ is isomorphic up to sign to $\tilde{\eta}_{M[m]}$.

Since the push-out of $\tilde{\eta}_{M[p^{r}]}$ along $G[p^{r}]\to G[p^{r+1}]$ is isomorphic to $p\cdot\tilde{\eta}_{M[p^{r+1}]}$, there is a morphism of complexes $\eta_{M[p^{r}]}\to\eta_{M[p^{r+1}]}$ for any $r\geq 1$. One then gets a short exact sequence of Barsotti -Tate groups (BT groups for short)

[TABLE]

where $L[p^{\infty}]:=L\otimes_{\mathbb{Z}}\mathbb{Q}_{p}/\mathbb{Z}_{p}$.

2.0.2 Lemma**.**

Let $M_{0}=[u_{0}\colon L_{0}\to G_{0}]$ and $\varphi_{0}\colon M_{0}\to N_{0}$ a morphism of $1$-motives over $k$.

1. (i)

Any finite and flat $R$-group scheme $M[m]$ which lifts $M_{0}[m]$ is naturally endowed with a weight filtration which lifts the weight filtration $T_{0}[m]\subseteq G_{0}[m]\subseteq M_{0}[m]$. If $M_{0}$ lifts to a $1$-motive $M$ over $R$, then the natural weight filtration on $M[m]$ agrees with the one induced by the weight filtration on $M$. 2. (ii)

Any morphism of finite flat $R$-group schemes $M[m]\to N[m]$ which lifts $\varphi_{0}[m]$ respects filtrations. 3. (iii)

Any BT group over $R$ which lifts $M_{0}[p^{\infty}]$ is naturally endowed with a weight filtration and any morphism of BT groups over $R$ which lifts $\varphi_{0}[p^{\infty}]$ respects filtrations.

Proof.

Let $L$ be the unique étale lifting of $L_{0}$ over $R$. Let $M[m]$ be a finite and flat $R$-group scheme lifting $M_{0}[m]$. Then $M[m]$ fits into a short exact sequence

[TABLE]

which lifts the sequence $\eta_{M_{0}[m]}$ for $M_{0}$ (cf. (2.2)). Indeed, there exists a unique lifting $f\colon M[m]\to L/mL$ of $M_{0}[m]\to L_{0}/mL_{0}$ since $L/mL$ is étale and we set $G[m]=\operatorname{Ker}f$. Note further that $f$ factors through the epimorphism $\pi_{M}\colon M[m]\to M[m]^{\rm\acute{e}t}=M[m]/M[m]^{\circ}$ with $M[m]^{\circ}$ the identity component of $M[m]$. Let $f^{\rm\acute{e}t}\colon M[m]^{\rm\acute{e}t}\to L/mL$ satisfy $f^{\rm\acute{e}t}\circ\pi_{M}=f$. Since $\pi_{M}$ is an fppf epimorphism, it suffices to prove that the same is $f^{\rm\acute{e}t}$. This follows immediately since $f^{\rm\acute{e}t}_{0}$ is an epimorphism of finite étale $k$-group schemes.

Note further that $M[m]$ is naturally filtered. Indeed, let $T$ be the unique torus lifting $T_{0}$, $L^{*}$ its group of characters and $M[m]^{*}$ the Cartier dual of $M[m]$. We have seen that the canonical morphism $M_{0}[m]^{*}\to L_{0}^{*}/mL^{*}_{0}$ lifts uniquely over $R$ and hence, by Cartier duality, the immersion $T_{0}[m]\to M_{0}[m]$ lifts uniquely over $R$. Now, the composition $T[m]\to M[m]\to L/mL$ is the zero map, since its reduction modulo $\mathfrak{m}$ is the zero map and $L/mL$ is étale. Hence the closed immersion $T[m]\to M[m]$ factors through $G[m]$ and (i) is proved.

With similar arguments one shows (ii). Finally (iii) follows immediately from (i) and (ii). ∎

The classical Serre -Tate theorem says that deformations of an abelian variety over $k$ only depend on deformations of its BT group. As we will now explain, the analogous result for $1$-motives of the type $[\mathbb{Z}\to\mathbb{G}_{m,k}]$ can explicitly be deduced from the canonical isomorphism

[TABLE]

where $x_{0}$ is the reduction of $x$ modulo $\mathfrak{m}$ and $[x_{0}]\in W_{s}(k)$ is the multiplicative representative of $x_{0}$. Recall that $(1+\mathfrak{m})^{p^{n}}=\{1\}$ and hence

[TABLE]

We now consider BT groups which are extensions of $\mathbb{Q}_{p}/\mathbb{Z}_{p}$ by $\mu_{p^{\infty}}$. By [8, Proposition 2.5 p. 180] there exists an isomorphism

[TABLE]

which associates to $a\in 1+\mathfrak{m}$ the push-out along $a\colon\mathbb{Z}\to\mu_{p^{\infty}}$ of the sequence

[TABLE]

Let $\mathrm{Ext}_{\mathbb{Z}/p^{n}\mathbb{Z}}(-,-)$ denote classes of extensions of $R$-group schemes killed by $p^{n}$.

2.0.3 Lemma**.**

There is an isomorphism of groups $\Psi_{n}\colon 1+\mathfrak{m}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Ext}_{\mathbb{Z}/p^{n}\mathbb{Z}}(\mathbb{Z}/p^{n}\mathbb{Z},\mu_{p^{n}})$ which associates to $a\in 1+\mathfrak{m}$ the short exact sequence $\Psi(a)[p^{n}]$, i.e., the short exact sequence of kernels of the multiplication by $p^{n}$ associated to $\Psi(a)$.

Proof.

By [8, p.183] we have isomorphisms

[TABLE]

where the first isomorphism is induced by pull-back along $\mathbb{Z}/p^{m}\mathbb{Z}\to\mathbb{Q}_{p}/\mathbb{Z}_{p}$, the second isomorphism follows since $p^{n}$ kills $\mathrm{Ext}_{R}(\mathbb{Q}_{p}/\mathbb{Z}_{p},\mu_{p^{\infty}})$, and the third isomorphism is obtained by passing to kernels of the multiplication by $p^{n}$. By definition $\Psi_{n}$ is the composition of the isomorphism $\Psi$ in (2.5) with the isomorphisms in (2.6). Hence the conclusion follows. ∎

2.0.4 Remark**.**

Note that $\Psi_{n}(a)$ is also represented by the sequence of kernels of the multiplication by $p^{n}$ of the sequence $\widetilde{\Psi}(a)$ obtaied via push-out along $a\colon\mathbb{Z}\to\mathbb{G}_{m,R}$ from $\zeta\colon 0\to\mathbb{Z}\to\mathbb{Z}\to\mathbb{Z}/p^{n}\mathbb{Z}\to 0$. Indeed, $\widetilde{\Psi}(a)=\iota_{n}^{*}\iota_{*}\Psi(a)$ with $\iota_{n}\colon\mathbb{Z}/p^{n}\mathbb{Z}\to\mathbb{Q}_{p}/\mathbb{Z}_{p}$ and $\iota\colon\mu_{p^{\infty}}\to\mathbb{G}_{m,R}$ the canonical morphisms.

We now see how to interpret (2.4) in terms of deformations of $1$-motives.

2.0.5 Lemma**.**

Given a $1$-motive $M_{0}=[u_{0}\colon\mathbb{Z}\to\mathbb{G}_{m,k}]$ and a $\mathcal{B}\in\mathrm{Ext}_{R}(\mathbb{Q}_{p}/\mathbb{Z}_{p},\mu_{p^{\infty}})$ there is a unique $1$-motive $M=[u\colon\mathbb{Z}\to\mathbb{G}_{m,R}]$ which lifts $M_{0}$ and whose BT group is isomorphic to $\mathcal{B}$ as extension of $\mu_{p^{\infty}}$ by $\mathbb{Q}_{p}/\mathbb{Z}_{p}$.

Proof.

Note that if $\mathcal{B},\mathcal{B}^{\prime}$ are two liftings of $M_{0}[p^{\infty}]$ there is at most one morphism $\mathcal{B}\to\mathcal{B}^{\prime}$ as extensions of $\mathbb{Q}_{p}/\mathbb{Z}_{p}$ by $\mu_{p^{\infty}}$.

We have to prove that the homomorphism

[TABLE]

which maps $u\in R^{*}$ to the pair $(u_{0},M^{u}[p^{\infty}])$, where $u_{0}$ is the reduction of $u$ modulo $\mathfrak{m}$ and $M^{u}=[u\colon\mathbb{Z}\to\mathbb{G}_{m,R}]$, is an isomorphism. Note that if $u=[u_{0}],u_{0}\in k^{*}$, then $u$ admits $p^{r}$th roots for any $r\geq 1$; hence $\eta_{M^{u}[p^{\infty}]}$ is split since by (2.5) $\mathrm{Ext}_{R}(\mathbb{Q}_{p}/\mathbb{Z}_{p},\mu_{p^{\infty}})$ is killed by $p^{n}$. We are then reduced to checking that

[TABLE]

is an isomorphism. Thanks to (2.6) we may view it as the homomorphism

[TABLE]

which maps $u$ to $\eta_{M^{u}[p^{n}]}$. Since $u$ is $p^{n}$-torsion, we may write it as a morphism $u\colon\mathbb{Z}/p^{n}\mathbb{Z}\to\mathbb{G}_{m,R}$. Let $u^{D}\colon\mathbb{Z}\to\mu_{p^{n}}$ be the morphism obtained by applying Cartier duality to $u$. By Remark 2.0.1 $\Phi_{n}(u)=\eta_{M^{u}[p^{n}]}$ is, up to sign, the sequence of cokernels of the multiplication by $p^{n}$ of the sequence obtained via pull-back along $u\colon\mathbb{Z}\to\mathbb{G}_{m,R}$ from the sequence $0\to\mu_{p^{n}}\to\mathbb{G}_{m,R}\to\mathbb{G}_{m,R}\to 0$. Passing to Cartier duals and recalling Remark 2.0.4, we then get that $\Phi(u)^{D}=-\Psi_{n}(u^{D})$. The conclusion follows now from Lemma 2.0.3 and the fact that Cartier duality induces isomorphisms on both $1+\mathfrak{m}=\operatorname{Hom}_{R-{\rm gr}}(\mathbb{Z}/p^{n}\mathbb{Z},\mu_{p^{n}})$ and $\mathrm{Ext}_{R}(\mathbb{Q}_{p}/\mathbb{Z}_{p},\mu_{p^{\infty}})$. ∎

3. Intermediate results

3.0.1 Remark**.**

We can refine some preliminary results in [6].

1. (a)

Let $G$ be any smooth commutative $R$-group scheme. By [6, Lemma 1.1.2] the kernel of the reduction map $G(R)\to G(k)$ is killed by $p^{s(s^{\prime}-1)}$. In our hypothesis one can prove that it is killed by $p^{s^{\prime}-1}$. Indeed, by the theory of Greenberg functor the sections $G(R/\mathfrak{m}^{i})$, $1\leq i\leq s^{\prime}$, can be identified with the group of $k$-rational sections of a smooth $k$-group scheme $\mathrm{Gr}_{i}(G)$. Further there are so-called change of level morphisms $\varrho_{i}^{1}\colon\mathrm{Gr}_{i+1}(G)\to\mathrm{Gr}_{i}(G)$ such that $\varrho_{i}^{1}(k)$ coincides with the reduction map $G(R/\mathfrak{m}^{i+1})\to G(R/\mathfrak{m}^{i})$. By Greenberg’s structure theorem ([5],[3, Thm. 12.13]) the kernel of $\varrho_{i}^{1}$ is a $k$-vector group, thus killed by $p$. It follows then by induction that $\operatorname{Ker}(G(R)\to G(k))$ is killed by $p^{s^{\prime}-1}$. 2. (b)

By [6, Lemma 1.1.3 (3)] a morphism of semi-abelian varieties $\varphi_{0}\colon G_{0}\to H_{0}$ lifts over $R$ up to multiplication by $p^{s(s^{\prime}-1)}$. We can improve also this estimate if $p>2$. Let $M$, $N$ be $1$-motives over $R$. By [1, Thm. 2.1] there exists a canonical morphism between universal extensions (2.1) $\varphi^{\natural}\colon M^{\natural}\to N^{\natural}$ which lifts $\varphi_{0}^{\natural}$. If $\varphi^{\natural}(\mathbb{V}(M))\subseteq\mathbb{V}(N)$, then $\varphi^{\natural}$ induces a morphism $\varphi\colon M\to N$ which lifts $\varphi_{0}$. In general, since the multiplication by $p^{s}$ morphism kills $\mathbb{V}(M)$, the morphism $p^{s}\varphi^{\natural}$ maps $\mathbb{V}(M)$ to [math]. Hence $p^{s}\varphi_{0}$ lifts to a morphism \text{``p^{s}\varphi''}\colon M\to N.

3.1. Galois actions

Let $k^{\prime}/k$ be a finite Galois extension of Galois group $\Gamma=\mathrm{Gal}(k^{\prime}/k)$ and set $R^{\prime}=R\otimes_{W(k)}W(k^{\prime})$. Note that $R^{\prime}$ is an artinian local ring with residue field $k^{\prime}$ and $\mathrm{Spec}\,R^{\prime}\to\mathrm{Spec}\,R$ is a Galois covering of group $\Gamma$. Then $\Gamma$ naturally acts on $\mathcal{M}_{1}(R^{\prime})$ and $\mathrm{Def}(R^{\prime},k^{\prime})$ and Serre -Tate functor $\Delta_{R^{\prime}}$ (1.1) commutes with the Galois action.

Note that the datum of a $1$-motive $M$ over $R$ is equivalent to the datum of a $1$-motive $M^{\prime}=[u^{\prime}\colon L^{\prime}\to G^{\prime}]$ over $R^{\prime}$ with a $\Gamma$-action compatible with the $\Gamma$-action on $\mathrm{Spec}\,R^{\prime}$. Indeed $L^{\prime}$ descends to a lattice over $R$. Further, since $G^{\prime}$ is a separated $R^{\prime}$-scheme homeomorphic to the semi-abelian $k^{\prime}$-variety $G^{\prime}_{0}$, it can be covered by affine open subschemes which are stable under the $\Gamma$-action and thus it descends to a $R$-group scheme. The maximal subtorus $T^{\prime}$ of $G^{\prime}$ descends to a subtorus $T$ of $G$ and $G/T$ is an abelian scheme since it is an abelian scheme after base change to $R^{\prime}$. Similarly, the datum of an object $(G_{0},\mathcal{B},\varepsilon_{0})$ in $\mathrm{Def}(R,k)$ is equivalent to the datum of a $(G^{\prime}_{0},\mathcal{B}^{\prime},\varepsilon^{\prime}_{0})$ in $\mathrm{Def}(R^{\prime},k^{\prime})$ together with a $\Gamma$-action compatible with the $\Gamma$-action on $k^{\prime}$ (in the first and third entries) and the $\Gamma$-action on $R^{\prime}$ (on the second entry).

3.2. Serre -Tate theorem for semi-abelian varieties

Recall that Cartier duality is a self duality on $\mathcal{M}_{1}(R)$ which provides a contravariant equivalence between the categories $\mathcal{M}_{1}(R)_{\leq-1}$ and $\mathcal{M}_{1}(R)_{\geq-1}$ [4, 10.2], [2, §1]. Further if $M^{*}$ is the Cartier dual of a $1$-motive $M$, $M^{*}[p^{\infty}]$ is the Cartier dual of the BT group $M[p^{\infty}]$.

Consider the functor

[TABLE]

induced by (1.1). The proof of the next theorem contains several arguments used for the proof of the more general Serre -Tate Theorem 1.0.1. The fact that (3.1) is an equivalence of categories is needed later to prove both the fullness and the essential surjectiveness of (1.1). For this reason we can not jump directly to the general case.

3.2.1 Theorem**.**

The functor (3.1) is an equivalence of categories.

Proof.

The faithfulness can be proved as in [6, p. 144] for the case of abelian varieties. Let us give an alternative proof when $p>2$. If $\varphi\colon G\to H$ is a morphism in $\mathcal{M}_{1}(R)_{\leq-1}$ and $\varphi_{0}=0$ then the induced morphism between universal extensions $\varphi^{\natural}\colon G^{\natural}\to H^{\natural}$ is the zero map [1, Thm. 2.1] and hence $\varphi=0$. We now prove the fullness when $p>2$; if $p=2$ the same proof works replacing $s$ with $s(s^{\prime}-1)$ and it coincides with the one in [6, p. 144]. Given a $\varphi_{0}\colon G_{0}\to H_{0}$ and a morphism $\psi\colon G[p^{\infty}]\to H[p^{\infty}]$ lifting $\varphi_{0}[p^{\infty}]$ by Remark 3.0.1 (b) there exists a morphism \text{``p^{s}\varphi''}\colon G\to H lifting $p^{s}\varphi_{0}$. Further p^{s}\psi=\text{``p^{s}\varphi''}[p^{\infty}] by [6, Lemma 1.1.3(2)] since both morphisms lift $p^{s}\varphi_{0}[p^{\infty}]$. Hence “$p^{s}\varphi$” kills $G[p^{s}]$ and thus \text{``p^{s}\varphi''}=p^{s}\varphi for a morphism $\varphi\colon G\to H$ (necessarily unique since $\mathrm{Hom}_{R\text{-gr}}(G,H)$ has no $p$-torsion). Thus the fully faithfulness is proved.

Let now $(G_{0},\mathcal{B},\varepsilon_{0})$ be an object in $\mathrm{Def}(R,k)_{\leq-1}$. If the maximal subtorus of $G_{0}$ is split of dimension $d$, then $G_{0}$ lifts to an $R$-group scheme $G$ which is extension of an abelian scheme by a $\mathbb{G}_{m,R}^{d}$ and, for proving the essential surjectivity, one can repeat word by word the proof of the classical Serre -Tate theorem [6, Thm. 1.2.1]. Indeed the Cartier dual of $G_{0}$ is a $1$-motive $[w_{0}\colon\mathbb{Z}^{d}\to A^{*}_{0}]$ with $A_{0}^{*}$ an abelian variety, which lifts to an abelian scheme $A^{*}$ over $R$. Hence $w_{0}$ lifts to a morphism $w\colon\mathbb{Z}^{d}\to A^{*}$ over $R$ and by applying again Cartier duality, one gets $G$.

For the general case, one proceeds by Galois descent as follows (cf. Subsection 3.1). Let $k^{\prime}/k$ be a finite Galois extension such that the maximal subtorus of $G_{0}$ becomes split over $k^{\prime}$. Let $\Gamma=\mathrm{Gal}(k^{\prime}/k)$ and set $R^{\prime}=R\otimes_{W(k)}W(k^{\prime})$. Let $(G^{\prime}_{0},\mathcal{B}^{\prime},\varepsilon^{\prime}_{0})$ in $\mathrm{Def}(R^{\prime},k^{\prime})_{\leq-1}$ be obtained from $(G_{0},\mathcal{B},\varepsilon_{0})$ via base change along $k^{\prime}/k$ (on semi-abelian varieties) and $R^{\prime}/R$ (on BT groups). For any $\sigma\in\Gamma$ we have an isomorphism in $\mathrm{Def}(R^{\prime},k^{\prime})_{\leq-1}$

[TABLE]

with

[TABLE]

By the previous step we know that there exists a $G^{\prime}$ in $\mathcal{M}_{1}(R^{\prime})_{\leq-1}$ whose image in $\mathrm{Def}(R^{\prime},k^{\prime})_{\leq-1}$ is the triple $(G^{\prime}_{0},\mathcal{B}^{\prime},\varepsilon^{\prime}_{0})$. By the fully faithfulness of (3.1) the isomorphism $(\varphi_{\sigma,0},\psi_{\sigma})$ gives a unique isomorphism $\varphi_{\sigma}\colon G^{\prime}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\sigma^{*}G^{\prime}$ which lifts $\varphi_{\sigma,0}$ and restricts to $\psi_{\sigma}$ on BT groups. These morphisms define a $\Gamma$-action on $G^{\prime}$ compatible with the action on $\mathrm{Spec}\,R^{\prime}$ and hence $G^{\prime}$ descends to a $G$ in $\mathcal{M}_{1}(R)_{\leq-1}$ with BT group isomorphic to $\mathcal{B}$. ∎

Theorem 3.2.1 allows another step towards the proof of Theorem 1.0.1.

3.2.2 Corollary**.**

The functor $\mathcal{M}_{1}(R)_{\geq-1}\to\mathrm{Def}(R,k)_{\geq-1}$ induced by (1.1) is an equivalence of categories.

Proof.

It is sufficient to apply Cartier duality to the previous result. ∎

4. The general case

4.1. The faithfulness

The proof of the faithfulness of (1.1) follows easily from faithfulness on $\mathcal{M}_{1}(R)_{\leq-1}$.

4.1.1 Proposition**.**

Let $M,N$ be two $1$-motives over $R$. Then the reduction map

[TABLE]

is injective.

Proof.

Consider a morphism $\varphi\colon M\to N$ and assume that its reduction modulo $\mathfrak{m}$ is the [math] morphism. Hence $\varphi=0$ in degree $-1$ by the equivalence of the étale sites over $R$ and over $k$ and $\varphi=0$ in degree [math] by Theorem 3.2.1. ∎

4.1.2 Corollary**.**

The functor (1.1) is faithful.

4.2. The fullness

For the proof we will need the following lemma.

4.2.1 Lemma**.**

Let $M=[u\colon L\to G]$ be a $1$-motive over $R$. If $u_{0}=0$ and $M[p^{\infty}]$ is split extension of $L[p^{\infty}]$ by $G[p^{\infty}]$, then $u=0$.

Proof.

We may work étale locally on $R$ and assume $L$ constant and $T$ a split torus. By Corollary 3.2.2 $u_{\mathrm{ab}}\colon L\to A$ is the [math] morphism. Hence $u$ factors through $T$. Let $M_{t}=[u\colon L\to T]$ and note that $M_{t}[p^{\infty}]$ is split extension $L[p^{\infty}]$ by $T[p^{\infty}]$. Hence, $u=0$ by Lemma 2.0.5. ∎

4.2.2 Proposition**.**

The functor (1.1) is full.

Proof.

Let $M=[u\colon L\to G],N=[v\colon F\to H]$ be two $1$-motives over $R$, $\varphi_{0}\colon M_{0}\to N_{0}$ a morphism between their reduction modulo $\mathfrak{m}$ and $\psi\colon M[p^{\infty}]\to N[p^{\infty}]$ a lifting of $\varphi_{0}[p^{\infty}]\colon M_{0}[p^{\infty}]\to N_{0}[p^{\infty}]$. We have to prove that there exists a lifting $\varphi\colon M\to N$ of $\varphi_{0}$ over $R$ such that $\varphi[p^{\infty}]=\psi$.

Let $\varphi_{0}=(f_{0},g_{0})$, i.e., $f_{0}\colon L_{0}\to F_{0},g_{0}\colon G_{0}\to H_{0}$ and $g_{0}\circ u_{0}=v_{0}\circ f_{0}$. Recall that any lifting $\psi$ of $\varphi_{0}[p^{\infty}]$ respects weight filtrations by Lemma 2.0.2. Hence, by Theorem 3.2.1 there exists a unique morphism $g\colon G\to H$ lifting $g_{0}$. Further, by the equivalence between the category of étale group schemes over $k$ and the category of étale group schemes over $R$, there exists a unique $f\colon L\to F$ lifting $f_{0}$. We are left to prove that $g\circ u=v\circ f$, so that $\varphi=(f,g)$ is a morphism of $1$-motives, and that $\varphi[p^{\infty}]=\psi$. The latter equality follows from [6, Lemma 1.1.3 2)] since both morphisms are liftings of $\varphi_{0}[p^{\infty}]$.

Let $Z=[w\colon L\to H]$ with $w=g\circ u-v\circ f$. We claim that $\eta_{Z[p^{\infty}]}$ in (2.3) is split. For proving this claim, it is sufficient to check that $\eta_{Z[p^{r}]}$ (and hence $\tilde{\eta}_{Z[p^{r}]}$ in (2.2)) is split for any $r$. Since $\psi[p^{r}]$ restricts to the morphism $g[p^{r}]\colon G[p^{r}]\to H[p^{r}]$ on weight $\leq-1$ subgroups and induces $f/p^{r}f\colon L/p^{r}L\to F/p^{r}F$ in weight [math], it is

[TABLE]

and hence

[TABLE]

We conclude then by applying Remark 2.0.1 that

[TABLE]

Hence $Z$ is a $1$-motive such that $\eta_{Z[p^{\infty}]}$ is split extension of $L[p^{\infty}]$ by $H[p^{\infty}]$ and $w_{0}=0$. Thus $w=0$ by Lemma 4.2.1. ∎

4.3. Essential surjectiveness

The strategy of the proof is first to construct the desired $1$-motive étale locally and then to apply descent. By Theorem 3.2.1, Corollary 3.2.2 and Lemma 2.0.5 we have already partial results in this direction. We prove another special case.

4.3.1 Lemma**.**

Let $M_{0}=[u_{0}\colon\mathbb{Z}^{m}\to G_{0}]$ be a $1$-motive over $k$ with $T_{0}=\mathbb{G}_{m,k}^{d}$ a split torus. Let $\mathcal{B}$ be a BT group over $R$ lifting $M_{0}[p^{\infty}]$. Then there exists a unique (up to unique isomorphism) $1$-motive $M$ over $R$ lifting $M_{0}$ and with BT group isomorphic to $\mathcal{B}$.

Proof.

Uniqueness follows by Proposition 4.1.1. For the existence, recall that by Lemma 2.0.2 $\mathcal{B}$ is naturally filtered so that $W_{-1}\mathcal{B}$ is a lifting of $G_{0}[p^{\infty}]$ and $\mathcal{B}/W_{-2}\mathcal{B}$ is a lifting of $M_{0,\mathrm{ab}}[p^{\infty}]$. By Theorem 3.2.1 $G_{0}$ lifts to an $R$-scheme $G$ which is extension of an abelian scheme $A$ by $T\simeq\mathbb{G}_{m,R}^{d}$ and $G[p^{\infty}]=W_{-1}\mathcal{B}$; further $M_{0,\mathrm{ab}}$ lifts to a $1$-motive $M_{A}=[u_{A}\colon\mathbb{Z}^{m}\to A]$ whose BT group is isomorphic to $\mathcal{B}/W_{-2}\mathcal{B}$ by Corollary 3.2.2.

Let $M^{\prime}=[u^{\prime}\colon\mathbb{Z}^{m}\to G]$ be any extension of $M_{A}$ by $T$; it exists since $H^{1}(R,\mathbb{G}_{m,R})=0$. Since $T(R)\to T(k)$ is surjective, we may assume that $u^{\prime}$ is also a lifting of $u_{0}$. We are then left to alter $u^{\prime}$ so that $M^{\prime}[p^{\infty}]\simeq\mathcal{B}[p^{\infty}]$ in $\operatorname{Ext}_{R}((\mathbb{Q}_{p}/Z_{p})^{m},G[p^{\infty}])$.

Let $\mathcal{E}=\mathcal{B}[p^{n}]-M^{\prime}[p^{n}]$ as extension of $(\mathbb{Z}/p^{n}\mathbb{Z})^{m}$ by $G[p^{n}]$. Since the push-out along $G[p^{n}]\to A[p^{n}]$ maps $\mathcal{E}$ to the trivial extension of $(\mathbb{Z}/p^{n}\mathbb{Z})^{m}$ by $A[p^{n}]$, there exists by Lemma 2.0.5 a morphism $v\colon\mathbb{Z}^{m}\to T$ such that $M=[u=u^{\prime}-v\colon\mathbb{Z}^{m}\to G]$ is a lifting of $M_{0}$ and $M[p^{n}]$ is isomorphic to $\mathcal{B}[p^{n}]$. Further $M[p^{\infty}]$ and $\mathcal{B}$ induce the same extension of $(\mathbb{Q}_{p}/\mathbb{Z}_{p})^{m}$ by $A[p^{\infty}]$. Hence they differ by an $\eta\in\mathrm{Ext}_{R}((\mathbb{Q}_{p}/\mathbb{Z}_{p})^{m},\mu_{p^{\infty}}^{d})$ which induces a split extension on the $p^{n}$-torsion subgroups. By (2.6) we conclude that $\eta=0$ and hence $M[p^{\infty}]\simeq\mathcal{B}$.∎

Let $k^{\prime}/k$ be a finite Galois extension with Galois group $\Gamma$ and set $R^{\prime}=R\otimes_{W(k)}W(k^{\prime})$. As remarked in Subsection 3.1, if a $1$-motive $M^{\prime}$ over $R^{\prime}$ descends to a $1$-motive $M$ over $R$, then $\Delta_{R^{\prime}}(M^{\prime})$ in $\mathrm{Def}(R^{\prime},k^{\prime})$ descends to $\Delta_{R}(M)$ in $\mathrm{Def}(R,k)$. The next lemma shows that also the converse holds.

4.3.2 Lemma**.**

Let notation be as above. A $1$-motive $M^{\prime}$ descends to $R$ if, and only if, its image in $\mathrm{Def}(R^{\prime},k^{\prime})$ descends to $\mathrm{Def}(R,k)$.

Proof.

Let $M^{\prime}=[u^{\prime}\colon L^{\prime}\to G^{\prime}]$ and assume that $(M^{\prime}_{0},\mathcal{B}^{\prime}=M^{\prime}[p^{\infty}],\mathrm{can}\colon\mathcal{B}^{\prime}_{0}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}M^{\prime}_{0}[p^{\infty}])$ descends to an object $(M_{0}=[L_{0}\to G_{0}],\mathcal{B},\varepsilon_{0}\colon\mathcal{B}_{0}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}M_{0}[p^{\infty}]))$ in $\mathrm{Def}(R,k)$. For any $\sigma\in\Gamma$ we then have an isomorphism in $\mathrm{Def}(R^{\prime},k^{\prime})$

[TABLE]

where

[TABLE]

make the following diagram

[TABLE]

commute. By the fully faithfulness of (1.1) the isomorphism $(\varphi_{\sigma,0},\psi_{\sigma})$ gives a unique isomorphism $\varphi_{\sigma}\colon M^{\prime}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\sigma^{*}M^{\prime}$ which lifts $\varphi_{\sigma,0}$ and restricts to $\psi_{\sigma}$ on BT groups. Hence we have defined an action of $\Gamma$ on $M^{\prime}$, in particular on $L^{\prime}$ and on $G^{\prime}$. Since $L^{\prime}$ is étale over $k^{\prime}$, it descends to a lattice $L$ whose special fiber is $L_{0}$. On the other hand, the restriction of the $\Gamma$-action on weight $\leq-1$ gives a $G$ in $\mathcal{M}_{1}(R)_{\leq-1}$ by Theorem 3.2.1. Finally, since $u^{\prime}$ is $\Gamma$-equivariant, it descends to a morphism $u\colon L\to G$. By construction now the $1$-motive $M^{\prime}$ descends to $M=[u\colon L\to G]$ and the image of $M$ in $\mathrm{Def}(R,k)$ is $(M_{0},\mathcal{B},\varepsilon_{0})$. ∎

4.3.3 Proposition**.**

The functor (1.1) is essentially surjective.

Proof.

Let $(M_{0}=[L_{0}\to G_{0}],\mathcal{B},\varepsilon_{0})$ be an object of $\mathrm{Def}(R,k)$. Thanks to Lemma 4.3.2 we may assume that $L_{0}$ is constant and that the maximal subtorus of $G_{0}$ is split. The conclusion follows then by Lemma 4.3.1. ∎

With this proposition the proof of Theorem 1.0.1 is completed. We should add that some results on deformations of $1$-motives were proved, with different methods, in Madapusi’s thesis [7], but were not published later.

Since any extension of an étale BT group by a toroidal BT group is split over $k$, we deduce the following generalization of Lemma 2.0.5.

4.3.4 Corollary**.**

Let $T$ be an $R$-torus and $L$ a lattice. For any $1$-motive $M_{0}=[u_{0}\colon L_{0}\to T_{0}]$ and any BT group $\mathcal{B}$ which is extension of $L[p^{\infty}]$ by $T[p^{\infty}]$ there is a unique $1$-motive $M=[u\colon L\to T]$ which lifts $M_{0}$ and whose BT group is isomorphic to $\mathcal{B}$.

4.4. Serre -Tate moduli for $1$-motives

Let now $k$ be an algebraically closed field of characteristic $p>0$. Following [6, § 2] one can define the formal moduli of a given $1$-motive $M_{0}$ over $k$. Namely let $\widehat{\mathscr{M}}_{M_{0}}=\widehat{\mathscr{M}}$ be the functor

[TABLE]

where $R$ is a local artin ring with residue field $k$. By Theorem 1.0.1 we get a bijection

[TABLE]

If $M_{0}=A_{0}$ is an ordinary abelian variety a construction of Serre -Tate gives an isomorphism of functors

[TABLE]

where $\widehat{\mathbb{G}}_{\rm m}(R)=1+\mathfrak{m}$ are the principal units of $R$ and $A^{*}_{0}$ is the dual abelian variety.

This result can be naturally extended to $1$-motives as follows.

4.4.1 Proposition**.**

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $M_{0}$ be a $1$-motive over $k$ such that $\operatorname{gr}_{-1}M_{0}=A_{0}$ is an ordinary abelian variety. Let $(R,\mathfrak{m})$ be an artinian local ring with residue field $k$.

• (i)

For any $1$-motive $M$ over $R$ lifting $M_{0}$ there exist a canonical $\mathbb{Z}_{p}$-bilinear form

[TABLE]

inducing an isomorphism of functors

[TABLE]

where $T_{p}M_{0}(k):=\varprojlim_{n}M_{0}[p^{n}](k)$.

• (ii)

Let $\varphi_{0}\colon M_{0}\to N_{0}$ be a morphism of $1$-motives over $k$ such that $\operatorname{gr}_{-1}M_{0},\operatorname{gr}_{-1}N_{0}$ are ordinary abelian varieties. Let $M$ and $N$ be liftings over $R$ of $M_{0}$ and $N_{0}$, respectively. Then $\varphi_{0}$ lifts to an $R$-morphism $\varphi\colon M\to N$ if, and only if,

[TABLE]

for every $\alpha\in T_{p}M_{0}(k)$ and $\beta^{*}\in T_{p}M_{0}^{*}(k)$. Further, is a lifting exists, it is unique.

Proof.

The proof goes exactly as in the classical case using the following two ingredients (cf. [6, p. 152]):

• •

the Poincaré biextension gives a perfect pairing $M[p^{n}]\times M^{*}[p^{n}]\to\mu_{p^{n}}$ inducing an isomorphism of functors

[TABLE]

since $M_{0}[p^{\infty}]^{\rm\acute{e}t}=T_{p}M_{0}(k)\otimes_{\mathbb{Z}_{p}}(\mathbb{Q}_{p}/\mathbb{Z}_{p})$. We denote by $E_{M}\colon M_{0}[p^{\infty}]^{\circ}\times T_{p}M_{0}(k)\to\widehat{\mathbb{G}}_{m}$ the corresponding pairing.

• •

Since $k$ is algebraically closed, the BT groups $M_{0}[p^{\infty}]^{\circ}$ and $M_{0}[p^{\infty}]^{\rm\acute{e}t}$ are split. Thus by [8, Proposition 2.5 p. 180]

[TABLE]

and there exists a unique $\phi_{M}\in\operatorname{Hom}_{\mathbb{Z}_{p}}(T_{p}M(k),M_{0}[p^{\infty}]^{\circ}(R))$ associated to the isomorphism class of the BT group $M_{0}[p^{\infty}]$.

Then we can define $q(M/R,\alpha,\beta):=E_{M}(\phi_{M}(\alpha),\beta)$ and follow word by word the proof in [6]. ∎