Serre-Tate for 1-motives

This post is an overview of the article arXiv:1704.01340 written with A. Bertapelle.

We generalise to 1-motives the classical correspondence between the following sets

• The set of deformations of an abelian variety \(A\) in characteristic \(p>0\);
• The set of deformations of the Barsotti-Tate group of \(A\).

A nice application is a concrete description of the formal moduli space of an ordinary abelian variety.

Deformation theory

Let \(X_0\) be a smooth scheme over a field \(k\) and let \((R,\frak m)\) be an Artin local ring with residue field \(k\). Deformation theory is the study of schemes \(X/R\) deforming \(X_0\), i.e. \(X_k\cong X_0\). Namely one looks for conditions under which such an \(X\) exists and how many of them we can find.

Assume (for simplicity) that \((\frak m)^2=(0)\) (i.e. we just deal with first order infinitesimal deformations). The basic result of Grothendieck on deformation theory tells that

there exists an obstruction \(o(X_0)\in H^2(X_0,T_{X_0/k}\otimes \frak m)\) such that \(o(X_0)=0\) iff there exists \(X/R\) deforming \(X_0\). if \(o(X_0)=0\) the set of deformations corresponds to \(H^1(X_0,T_{X_0/k}\otimes \frak m)\), and \(H^0(X_0,T_{X_0/k}\otimes \frak m)\) corresponds to the group \(Aut(X/X_0)\). It follows quite easily that we can deform (at least formally) an abelian scheme \(A_0\) as a scheme. Then because of properness and the existence of a zero section one can show that the group structure lifts too (with some work as explained in [4]).

Barsotti-Tate groups

The theory Barsotti-Tate groups (aka p-divisible groups) have many interesting applications like in Faltings’ proof of Mordell-Weil or the Fontaine’s results on the existence of abelian schemes over number rings. The motivating example is the following: let \(E/k\) be an elliptic curve over some field. Then the scheme-theoretic kernel \(E(\ell^n)\) of the multiplication by \(\ell^n\) is a finite flat group scheme of rank \(2\cdot \ell^n\). Now if \(\ell \ne char\ k\) we an recover this group scheme by the associated galois module \(E(\ell^n)(k^{sep})\cong (\mathbb{Z}/\ell^n\mathbb{Z})^2\). Whereas if \(\ell = p\) is the characteristic of our base field \(E(\ell^n)(k^{sep})\) is equal to \(\mathbb{Z}/\ell^n\mathbb{Z}\) or \(0\). This motivates the use of \(E(p^\infty)=\cup E(p^n)\) which is the basic example of a Barsotti-Tate group.

For the record the fancy definition is a s follows (\(p\) is fixed). A BT group over \(S\) (any base scheme) is an abelian fppf sheaf \(G\) over \(S\) such that

\(G\) is p-divisible \(G\) is p-torsion \(G[p]\) is a finite flat group scheme The main result

We fix \(k\) of characteristic \(p>0\) and we denote by \(Def(X_0/R)\) the set of deformations of \(X_0\) (a scheme, an abelian scheme or a BT group). Then Serre-Tate theory gives an isomorphism

for \(A_0\) an abelian scheme over \(k\). We proved the same statement (and more precisely a categorical equivalence) for \(A_0\) replaced by a 1-motve \(M_0\) over \(k\).

As a consequence if \(M_0\) is an ordinary 1-motive over an algebraically closed field we have the formula

and we can compute the Gauss-Manin connection of a deformation \(M/R\).

About the proofs

I don’t know the original proof given by Serre and Tate. In [1], which is the standard reference on the topic, is given the proof of Drinfel’d. Some part of it can be adapted to 1-motives, in particular for the fully faithfulness in weight \(\le 1\). The difficult part of the proof is the essential surjectivity. For instance one cannot use the Grothendieck deformation theorem in order to lift a general 1-motives: already for a semi-abelian scheme there is something to do!

Certainly the key point is to understand what happen for the 1-motive \([ \mathbb{Z} \to \mathbb{G}_m ]\). The BT group of such a 1-motive (say over \(\bar{\mathbb{F}}_p\)) is simply \(\mu_{p^\infty} \times \mathbb{Q}_p/\mathbb{Z}_p\) (like for an ordinary elliptic curve). By Kummer theory (see [2]) one gets

and I invite the interested reader to spend some time on this formula!

Further readings

I first learned about Serre-Tate theory by studying the personal notes of Olivier Brinon on BT groups. Then I went trough the classical references [1,2] with the help of [3,4,5].

[1] Katz: Serre-Tate local moduli

[2] Messing: Thesis, LNM 264

[3] Ward: blog posts ST1 and ST2

[4] Dospinescu: theorems of Serre-Tate and Grothendieck

[5] Amy (Cheung) Wooding: beamer on ST

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