2 Formal groups and Honda theory

Let $A$ be a ring. We denote by $X$ the $d$ -tuple of independent variables $(x_{1},\ldots ,x_{d})$ and write $X^{q},q\in \mathbb{N}$ , for the $d$ -tuple $(x_{1}^{q},\ldots ,x_{d}^{q})$ . We also consider the ring $A[[X]]_{0}$ of formal power series over $A$ in variables $x_{1},\ldots ,x_{d}$ without constant term.

A $d$ -dimensional formal group over $A$ is a $d$ -tuple of formal power series $F\in A[[X,Y]]^{d}$ satisfying the following properties

1. (i) $F(X,0)=X$ ;

2. (ii) $F(X,F(Y,Z))=F(F(X,Y),Z)$ ;

3. (iii) $F(X,Y)=F(Y,X)$ .

Let $F$ and $F^{\prime }$ be $d$ - and $d^{\prime }$ -dimensional formal groups over $A$ . A $d^{\prime }$ -tuple of formal power series $g\in A[[X]]_{0}^{d^{\prime }}$ is called a homomorphism from $F$ to $F^{\prime }$ , if $g(F(X,Y))=F^{\prime }(g(X),g(Y))$ . The matrix $D\in \text{M}_{d^{\prime },d}(A)$ satisfying $g(X)\equiv DX\hspace{0.6em}{\rm mod}\hspace{0.2em}\deg 2$ is called the linear coefficient of $g$ . It is easy to see that a homomorphism $g$ is an isomorphism if and only if its linear coefficient is an invertible matrix. A homomorphism with identity linear coefficient is called a strict isomorphism.

If $A$ is a $\mathbb{Q}$ -algebra, then for any $d$ -dimensional formal group $F$ over $A$ , there exists a unique strict isomorphism $f$ from $F$ to $F_{a}^{d}(X,Y)=X+Y$ (see for instance [Reference Honda10, Theorem 1]). This $f$ is called the logarithm of $F$ . It determines the formal group $F$ uniquely, since $F(X,Y)=f^{-1}(f(X)+f(Y))$ .

If $A$ is a ring of characteristic $0$ and $F$ is a formal group over $A$ , the logarithm of $F$ is by definition the logarithm of $F_{A\otimes \mathbb{Q}}$ . If, in addition, $F^{\prime }$ is another formal group over $A$ and $g$ is a homomorphism from $F$ to $F^{\prime }$ with linear coefficient $D$ , then $g={f^{\prime }}^{-1}\circ (Df)$ , where $f$ , $f^{\prime }$ are the logarithms of $F$ , $F^{\prime }$ , respectively (see [Reference Honda10, Proposition 1.6]).

Let $k$ be a perfect field of characteristic $p\neq 0$ , ${\mathcal{O}}$ be the ring of Witt vectors over $k$ , ${\mathcal{K}}$ be the quotient field of ${\mathcal{O}}$ , and $\unicode[STIX]{x1D6E5}:{\mathcal{K}}\rightarrow {\mathcal{K}}$ be the Frobenius automorphism. Let $E={\mathcal{O}}[[\blacktriangle ]]$ denote the ${\mathcal{O}}$ -algebra of noncommutative formal power series in the variable $\blacktriangle$ over ${\mathcal{O}}$ with the multiplication rule $\blacktriangle a=a^{\unicode[STIX]{x1D6E5}}\blacktriangle$ for any $a\in {\mathcal{O}}$ . Extend the ${\mathcal{O}}$ -module structure on ${\mathcal{K}}[[X]]_{0}$ to a left $E$ -module structure by the formula $\blacktriangle f(X)=f^{\unicode[STIX]{x1D6E5}}(X^{p})$ . It determines a bilinear map $\text{M}_{d^{\prime },d}(E)\times {\mathcal{K}}[[X]]_{0}^{d}\rightarrow {\mathcal{K}}[[X]]_{0}^{d^{\prime }}$ . In particular, it gives a left $\text{M}_{d}(E)$ -module structure on ${\mathcal{K}}[[X]]_{0}^{d}$ .

Denote $\mathfrak{P}=p{\mathcal{O}}[[X]]_{0}^{d}$ . Let $u\in \text{M}_{d}(E)$ be such that $u\equiv pI\hspace{0.6em}{\rm mod}\hspace{0.2em}\blacktriangle$ . We say that $u$ is a type of $f\in {\mathcal{K}}[[X]]_{0}^{d}$ if $f(X)\equiv X\hspace{0.6em}{\rm mod}\hspace{0.2em}\deg 2$ and $uf\in \mathfrak{P}$ .

The next technical result will be employed in what follows.

Honda theory [Reference Honda10] describes the logarithms of formal groups over ${\mathcal{O}}$ as formal power series of certain Honda type.

With the aid of Honda theory, the category of formal groups over $k$ can be also described and the Dieudonné module can be constructed. Under reduction of a formal power series with coefficients in ${\mathcal{O}}$ , we always mean the reduction modulo $p$ .

Let $F$ be a $d$ -dimensional formal group over ${\mathcal{O}}$ with logarithm $f\in {\mathcal{K}}[[X]]_{0}^{d}$ of type $u\in \text{M}_{d}(E)$ , and $\bar{F}$ denote the reduction of $F$ . The $E$ -module ${\mathcal{D}}(\bar{F})=\text{M}_{1,d}(E)f/\text{M}_{1,d}(E)f\cap p{\mathcal{O}}[[X]]_{0}$ is called the Dieudonné module of $\bar{F}$ . Proposition 2.3 implies that ${\mathcal{D}}(\bar{F})$ depends only on $\bar{F}$ . Since $uf\in \mathfrak{P}$ , the $E$ -linear map $\text{M}_{1,d}(E)\rightarrow {\mathcal{D}}(\bar{F})$ defined by $s\mapsto sf+p{\mathcal{O}}[[X]]_{0}$ induces a homomorphism $\text{M}_{1,d}(E)/\text{M}_{1,d}(E)u\rightarrow {\mathcal{D}}(\bar{F})$ which is an isomorphism by Proposition 2.4. A formal group $\bar{F}$ is said to be of finite height if ${\mathcal{D}}(\bar{F})$ is a free ${\mathcal{O}}$ -module of finite rank (see [Reference Fontaine7, Proposition III.6.1]). In this case the rank of ${\mathcal{D}}(\bar{F})$ is called the height of $\bar{F}$ .

The following theorem describes $\text{Aut}_{k}(\bar{F})$ in terms of Honda types.

Let ${\mathcal{K}}[[t]]_{p}=\{\sum _{i=0}^{\infty }b_{i}t^{p^{i}}|b_{i}\in {\mathcal{K}}\}$ and ${\mathcal{K}}[[X]]_{p}=\{\sum _{j=1}^{d}f_{j}(x_{j})|f_{j}\in {\mathcal{K}}[[t]]_{p}\}$ . For any $\sum _{i=0}^{\infty }a_{i}\blacktriangle ^{i}\in E$ , define $t^{p^{j}}\sum _{i=0}^{\infty }a_{i}\blacktriangle ^{i}=\sum _{i=0}^{\infty }a_{i}^{\unicode[STIX]{x1D6E5}^{j}}t^{p^{i+j}}$ . This provides to ${\mathcal{K}}[[t]]_{p}$ a right $E$ -module structure which in general looks as follows:

$$\begin{eqnarray}\left(\mathop{\sum }_{j=0}^{\infty }b_{j}t^{p^{j}}\right)\left(\mathop{\sum }_{i=0}^{\infty }a_{i}\blacktriangle ^{i}\right)=\mathop{\sum }_{i=0}^{\infty }\left(\mathop{\sum }_{j=0}^{i}b_{j}a_{i-j}^{\unicode[STIX]{x1D6E5}^{j}}\right)t^{p^{i}}.\end{eqnarray}$$

It determines a right $\text{M}_{d}(E)$ -module structure on ${\mathcal{K}}[[X]]_{p}^{d}$ . In fact, associating $\sum _{i=0}^{\infty }b_{i}t^{p^{i}}$ with $\sum _{i=0}^{\infty }b_{i}\blacktriangle ^{i}$ identifies ${\mathcal{K}}[[t]]_{p}$ with ${\mathcal{K}}[[\blacktriangle ]]$ provided with screw multiplication $\blacktriangle a=a^{\unicode[STIX]{x1D6E5}}\blacktriangle$ . Then the right action of $E$ on ${\mathcal{K}}[[t]]_{p}$ corresponds to the right multiplication in ${\mathcal{K}}[[\blacktriangle ]]$ . Similarly, ${\mathcal{K}}[[X]]_{p}^{d}$ and $\text{M}_{d}({\mathcal{K}}[[\blacktriangle ]])$ can be identified, and the right action of $\text{M}_{d}(E)$ on ${\mathcal{K}}[[X]]_{p}^{d}$ corresponds to the right multiplication in $\text{M}_{d}({\mathcal{K}}[[\blacktriangle ]])$ .

Covariant Honda theory [Reference Demchenko3] gives an alternative description of the category of $p$ -typical formal groups over ${\mathcal{O}}$ and can be considered as dual to Honda theory. Only one result from this theory will be used hereafter.

3 Universal $p$ -typical formal group and universal deformation

We consider polynomial rings with integer coefficients in an infinite number of variables which are grouped together in matrices for convenience. Let $V_{n}$ be $d\times d$ -matrices of independent variables $V_{n}(i,j)$ . The set of these independent variables is denoted by $V=(V_{1},V_{2},\ldots )$ .

According to Hazewinkel’s functional equation lemma [Reference Hazewinkel9, Section 10.2], the $d$ -tuple of power series $f_{V}\in \mathbb{Q}[V][[X]]_{0}^{d}$ defined by the recursion formula

$$\begin{eqnarray}f_{V}(X)=X+p^{-1}\mathop{\sum }_{n=1}^{\infty }V_{n}\unicode[STIX]{x1D70E}_{\ast }^{n}f_{V}(X^{p^{n}}),\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}(V_{n}(i,j))=V_{n}(i,j)^{p}$ , is the logarithm of a $d$ -dimensional formal group $F_{V}$ defined over $\mathbb{Z}[V]$ .

Let $a_{n}(V)\in \mathbb{Q}[V]$ denote the coefficients of $f_{V}$ , that means $f_{V}(X)=\sum _{n=0}^{\infty }a_{n}(V)X^{p^{n}}$ . It is clear that $a_{0}(V)=I$ . The following recurrence relation will be employed hereafter.

For a ring $A$ , denote by $A^{\infty }$ the $A$ -module consisting of the infinite sequences of elements of $A$ . Then $A_{m}^{\infty }$ is the submodule of $A^{\infty }$ consisting of the sequences starting with $m$ zeros.

A formal group $F$ over a ring $A$ is called $p$ -typical, if there exists $\unicode[STIX]{x1D6EF}:\mathbb{Z}[V]\rightarrow A$ such that $\unicode[STIX]{x1D6EF}_{\ast }F_{V}=F$ . A morphism $\unicode[STIX]{x1D6EF}:\mathbb{Z}[V]\rightarrow A$ is uniquely defined by the sequence $(\unicode[STIX]{x1D6EF}(V_{1}),\unicode[STIX]{x1D6EF}(V_{2}),\ldots )\in \text{M}_{d}(A)^{\infty }$ where $\unicode[STIX]{x1D6EF}(V_{n})\in \text{M}_{d}(A)$ is defined by $(\unicode[STIX]{x1D6EF}(V_{n}))(i,j)=\unicode[STIX]{x1D6EF}(V_{n}(i,j))$ . Thus we can write by abuse of the notation $\unicode[STIX]{x1D6EF}=(\unicode[STIX]{x1D6EF}(V_{1}),\unicode[STIX]{x1D6EF}(V_{2}),\ldots )$ . We will also write $F_{V(\unicode[STIX]{x1D6EF})}$ instead of $\unicode[STIX]{x1D6EF}_{\ast }F_{V}$ .

Let $k$ , ${\mathcal{O}}$ , $E$ be as in Section 2. Let $\unicode[STIX]{x1D6F7}$ be a $d$ -dimensional formal group over $k$ of height $h$ . According to [Reference Hazewinkel9, Corollary 25.4.29], $\unicode[STIX]{x1D6F7}$ is isomorphic to a $p$ -typical formal group over $k$ . Thus without loss of generality we can suppose that $\unicode[STIX]{x1D6F7}$ is $p$ -typical, i.e., there exists $\unicode[STIX]{x1D6EF}=(\unicode[STIX]{x1D6EF}_{1},\unicode[STIX]{x1D6EF}_{2},\ldots )\in \text{M}_{d}(k)^{\infty }$ such that $\unicode[STIX]{x1D6F7}=F_{V(\unicode[STIX]{x1D6EF})}$ . Let $\widehat{\unicode[STIX]{x1D6EF}}=(\widehat{\unicode[STIX]{x1D6EF}}_{1},\widehat{\unicode[STIX]{x1D6EF}}_{2},\ldots )\in \text{M}_{d}({\mathcal{O}})^{\infty }$ be the sequence of matrices which are composed of the multiplicative representatives in ${\mathcal{O}}$ of the entries of the matrices $\unicode[STIX]{x1D6EF}_{1},\unicode[STIX]{x1D6EF}_{2},\ldots$ Then $\widehat{F}=F_{V(\widehat{\unicode[STIX]{x1D6EF}})}$ is a formal group over ${\mathcal{O}}$ , and its reduction is equal to $\unicode[STIX]{x1D6F7}$ .

Given a sequence $T=(T_{1},T_{2},\ldots )\in \text{M}_{d}(k)^{\infty }$ , we introduce

$$\begin{eqnarray}Y_{n}(\unicode[STIX]{x1D6EF},T)=T_{1}\unicode[STIX]{x1D6EF}_{n-1}^{(p)}+\cdots +T_{n-1}\unicode[STIX]{x1D6EF}_{1}^{(p^{n-1})}\in \text{M}_{d}(k)\end{eqnarray}$$

for $n\geqslant 2$ and $Y_{1}(\unicode[STIX]{x1D6EF},T)=0$ . Then $Y_{\unicode[STIX]{x1D6EF}}(T)=(Y_{1}(\unicode[STIX]{x1D6EF},T),Y_{2}(\unicode[STIX]{x1D6EF},T),\ldots )$ defines a $k$ -linear operator on $\text{M}_{d}(k)^{\infty }$ .

For $n\geqslant 1$ , $1\leqslant i,j\leqslant d$ , denote by $B_{(n,i,j)}\in \text{M}_{d}(k)^{\infty }$ the sequence of matrices with the only nonzero entry which is equal to 1 and appears in the $n$ th matrix at the $(i,j)$ th position. According to [Reference Demchenko and Gurevich5, Proposition 8], there exists a set

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}\subset \{(n,i,j)|n\geqslant 1,1\leqslant i,j\leqslant d,B_{(n,i,j)}\notin \text{Im}Y_{\unicode[STIX]{x1D6EF}}\}\end{eqnarray}$$

such that $\{B_{\unicode[STIX]{x1D713}}+\text{Im}Y_{\unicode[STIX]{x1D6EF}}|\unicode[STIX]{x1D713}\in \unicode[STIX]{x1D6F9}\}$ is a basis of $\text{Coker}Y_{\unicode[STIX]{x1D6EF}}$ . Moreover, $|\unicode[STIX]{x1D6F9}|=dh$ .

Since $Y_{1}(V,T)=0$ , one has $\text{Im}Y_{\unicode[STIX]{x1D6EF}}\subset \text{M}_{d}(k)_{1}^{\infty }$ , and hence $\{(1,i,j)|1\leqslant i,j\leqslant d\}\subset \unicode[STIX]{x1D6F9}$ . Define $\unicode[STIX]{x1D6F9}^{+}=\{(n,i,j)|(n+1,i,j)\in \unicode[STIX]{x1D6F9}\}$ . Clearly, $|\unicode[STIX]{x1D6F9}^{+}|=d(h-d)$ .

Let $R$ be a complete Noetherian local ring with residue field $k$ , and $\unicode[STIX]{x1D6F7}$ be a formal group over $k$ . A formal group $F$ over $R$ with reduction $\unicode[STIX]{x1D6F7}$ is called a deformation of $\unicode[STIX]{x1D6F7}$ over $R$ . Let $F,F^{\prime }$ be deformations of $\unicode[STIX]{x1D6F7}$ over $R$ . An isomorphism from $F$ to $F^{\prime }$ with identity reduction is called a $\star$ -isomorphism.

Similarly to $B_{(n,i,j)}$ , denote by $\widehat{B}_{(n,i,j)}\in \text{M}_{d}(R)^{\infty }$ the sequence of matrices with the only nonzero entry being equal to 1 and appearing in the $n$ th matrix at the $(i,j)$ th position.

Now we can formulate the main result of [Reference Demchenko and Gurevich5].

4 Period map for deformations over ${\mathcal{O}}$

We keep the notation of the previous section. First, we prove an auxiliary result needed for the definition of the period map.

Denote $\widehat{f}=f_{V(\widehat{\unicode[STIX]{x1D6EF}})}$ . Clearly, $\widehat{f}$ is the logarithm of $\widehat{F}$ . Further, denote $\widehat{u}=pI-\sum _{n=1}^{\infty }\widehat{\unicode[STIX]{x1D6EF}}_{n}\blacktriangle ^{n}\in \text{M}_{d}(E)$ .

Let $F$ be a deformation of $\unicode[STIX]{x1D6F7}$ over ${\mathcal{O}}$ with logarithm $f$ . We denote the strict $\star$ -isomorphism classes of $F$ by $[F]$ , and the $\star$ -isomorphism classes of $F$ by $[F]^{\prime }$ . By Proposition 2.3, there is $v\in I+\blacktriangle \text{M}_{d}(E)$ such that $f\equiv v\widehat{f}\hspace{0.6em}{\rm mod}\hspace{0.2em}\mathfrak{P}$ . Then one can define the map $\unicode[STIX]{x1D712}$ from the set of the strict $\star$ -isomorphism classes of deformations of $\unicode[STIX]{x1D6F7}$ over ${\mathcal{O}}$ to

$$\begin{eqnarray}I+\text{M}_{d}(p{\mathcal{O}}+\blacktriangle E)/I+\text{M}_{d}(E)\widehat{u}\end{eqnarray}$$

and the map $\unicode[STIX]{x1D712}^{\prime }$ from the set of the $\star$ -isomorphism classes of deformations of $\unicode[STIX]{x1D6F7}$ over ${\mathcal{O}}$ to

$$\begin{eqnarray}I+\text{M}_{d}(p{\mathcal{O}})\backslash I+\text{M}_{d}(p{\mathcal{O}}+\blacktriangle E)/I+\text{M}_{d}(E)\widehat{u}\end{eqnarray}$$

as follows: $\unicode[STIX]{x1D712}([F])=v(I+\,\text{M}_{d}(E)\widehat{u})$ , $\unicode[STIX]{x1D712}^{\prime }([F]^{\prime })=(I+\text{M}_{d}(p{\mathcal{O}}))v(I+\,\text{M}_{d}(E)\widehat{u})$ .

One can see that $\unicode[STIX]{x1D712}$ and $\unicode[STIX]{x1D712}^{\prime }$ describe deformations of $\unicode[STIX]{x1D6F7}$ over ${\mathcal{O}}$ up to strict $\star$ -isomorphism and up to $\star$ -isomorphism, respectively. Theorem 3.2 provides an alternative description of the same deformations. Our aim is to establish an explicit connection between these two descriptions.

For $Z=(Z_{1},Z_{2},\ldots )\in \text{M}_{d}({\mathcal{O}})^{\infty }$ such that $Z\equiv \widehat{\unicode[STIX]{x1D6EF}}\hspace{0.6em}{\rm mod}\hspace{0.2em}p$ , we define

$$\begin{eqnarray}v_{Z}=I+p^{-1}\mathop{\sum }_{n=1}^{\infty }\mathop{\sum }_{k=1}^{n}a_{n-k}(Z)\left(Z_{k}^{(p^{n-k})}-\widehat{\unicode[STIX]{x1D6EF}}_{k}^{(p^{n-k})}\right)\blacktriangle ^{n}.\end{eqnarray}$$

Since $Z\equiv \widehat{\unicode[STIX]{x1D6EF}}\hspace{0.6em}{\rm mod}\hspace{0.2em}p$ , we conclude that $Z^{(p^{m})}\equiv \widehat{\unicode[STIX]{x1D6EF}}^{(p^{m})}\;\text{mod}\,p^{m+1}$ for any $m\geqslant 0$ . Moreover, Corollary of Lemma 3.1 implies $p^{m}a_{m}(Z)\in \text{M}_{d}({\mathcal{O}})^{\infty }$ for any $m\geqslant 0$ . Therefore $a_{n-k}(Z)\left(Z_{k}^{(p^{n-k})}-\widehat{\unicode[STIX]{x1D6EF}}_{k}^{(p^{n-k})}\right)\in \text{M}_{d}(p{\mathcal{O}})$ for any $n\geqslant k$ , and hence $v_{Z}\in \text{GL}_{d}(E)$ .

Proof. Since $\widehat{\unicode[STIX]{x1D6EF}}_{n}^{(p)}=\widehat{\unicode[STIX]{x1D6EF}}_{n}^{\unicode[STIX]{x1D6E5}}$ for any $n\geqslant 1$ , the recurrence formula of Lemma 3.1 can be rewritten as follows

$$\begin{eqnarray}pa_{n}(Z)-\mathop{\sum }_{k=1}^{n}a_{n-k}(Z)\widehat{\unicode[STIX]{x1D6EF}}_{k}^{\unicode[STIX]{x1D6E5}^{n-k}}=\mathop{\sum }_{k=1}^{n}a_{n-k}(Z)\left(Z_{k}^{(p^{n-k})}-\widehat{\unicode[STIX]{x1D6EF}}_{k}^{(p^{n-k})}\right).\end{eqnarray}$$

Then according to Proposition 2.6 and Lemma 4.2, one gets

$$\begin{eqnarray}f_{V(Z)}\widehat{u}=\left(\mathop{\sum }_{n=0}^{\infty }a_{n}(Z)X^{p^{n}}\right)\!\left(pI-\mathop{\sum }_{k=1}^{\infty }\widehat{\unicode[STIX]{x1D6EF}}_{k}\blacktriangle ^{k}\right)\!=v_{Z}(pX)=v_{Z}(\widehat{f}\widehat{u})=(v_{Z}\widehat{f})\widehat{u}.\end{eqnarray}$$

Since $\widehat{u}\equiv pI\hspace{0.6em}{\rm mod}\hspace{0.2em}\blacktriangle$ and the right multiplication by $\widehat{u}$ in ${\mathcal{K}}[[X]]_{p}^{d}$ corresponds to the right multiplication by $\widehat{u}$ in $\text{M}_{d}({\mathcal{K}}[[\blacktriangle ]])$ , we deduce that it is injective, and hence $f_{V(Z)}=v_{Z}\widehat{f}$ . This implies (1). Clearly, (2) follows from (1).◻

We have a natural bijection from the set $I+\text{M}_{d}(p{\mathcal{O}}+\blacktriangle E)/I+\text{M}_{d}(E)\widehat{u}$ to the set $\text{M}_{d}(p{\mathcal{O}}+\blacktriangle E)/\text{M}_{d}(E)\widehat{u}$ which maps $v(I+\text{M}_{d}(E)\widehat{u})$ to $v-I+\text{M}_{d}(E)\widehat{u}$ . The correspondence $pcI+\blacktriangle q+\text{M}_{d}(E)\widehat{u}\mapsto (pcI-cu)\blacktriangle ^{-1}+q+\text{M}_{d}(E)\widehat{u}$ is an ${\mathcal{O}}$ -module isomorphism from $\text{M}_{d}(p{\mathcal{O}}+\blacktriangle E)/\text{M}_{d}(E)\widehat{u}$ to $\text{M}_{d}(E)/\text{M}_{d}(E)\widehat{u}$ . The latter module is isomorphic as an ${\mathcal{O}}$ -module to the direct sum of $d$ copies of ${\mathcal{D}}(\unicode[STIX]{x1D6F7})$ . Therefore $\text{M}_{d}(p{\mathcal{O}}+\blacktriangle E)/\text{M}_{d}(E)\widehat{u}$ is a free ${\mathcal{O}}$ -module of rank $dh$ . Fix an ${\mathcal{O}}$ -basis of $\text{M}_{d}(p{\mathcal{O}}+\blacktriangle E)/\text{M}_{d}(E)\widehat{u}$ . Taking into account that $\unicode[STIX]{x1D712}$ is bijective (Proposition 4.3), one obtains a parametrization of the set of strict $\star$ -isomorphism classes of deformations of $\unicode[STIX]{x1D6F7}$ over ${\mathcal{O}}$ . Another parametrization of the same set is provided by Theorem 3.2(1) applied for $R={\mathcal{O}}$ . Using Proposition 4.4(1) and the arithmetic properties of the ring $\text{M}_{d}(E)$ , one can give an explicit formula which expresses the first parametrization through the second one. The same applies to two parameterizations of the set of $\star$ -isomorphism classes of deformations, one constructed with the aid of $\unicode[STIX]{x1D712}^{\prime }$ and the other coming from Theorem 3.2(2).

We illustrate the last observation for the set of $\star$ -isomorphism classes of deformations in a particular example where $\unicode[STIX]{x1D6EF}_{m}=I$ , for some $m\geqslant 1$ , $\unicode[STIX]{x1D6EF}_{n}=0$ for $n\neq m$ . This example is a direct generalization of the setting considered by Gross and Hopkins [Reference Gross and Hopkins8] for one-dimensional formal groups. In this case, one has $\widehat{u}=pI-I\blacktriangle ^{m}$ and the height of $\unicode[STIX]{x1D6F7}$ is equal to $dm$ . Moreover, $\unicode[STIX]{x1D6F9}=\{(n,i,j)|1\leqslant i,j\leqslant d;1\leqslant n\leqslant m\}$ (see [Reference Demchenko and Gurevich5, Example 1]), and hence, $\unicode[STIX]{x1D6F9}^{+}=\{(n,i,j)|1\leqslant i,j\leqslant d;1\leqslant n\leqslant m-1\}$ .

For any $(\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D713}})_{\unicode[STIX]{x1D713}\in \unicode[STIX]{x1D6F9}^{+}}$ , $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D713}}\in p{\mathcal{O}}$ , denote by $C_{n}$ the matrix in $\text{M}_{d}(p{\mathcal{O}})$ such that $C_{n}(i,j)=\unicode[STIX]{x1D70F}_{(n,i,j)}$ , $(n,i,j)\in \unicode[STIX]{x1D6F9}^{+}$ . If $Z=\widehat{\unicode[STIX]{x1D6EF}}+\sum _{\unicode[STIX]{x1D713}\in \unicode[STIX]{x1D6F9}^{+}}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D713}}\widehat{B}_{\unicode[STIX]{x1D713}}$ , then $Z_{n}=C_{n}$ for $1\leqslant n\leqslant m-1$ , $Z_{m}=I$ , $Z_{n}=0$ for $n\geqslant m+1$ , and

$$\begin{eqnarray}v_{Z}=I+p^{-1}\mathop{\sum }_{n=1}^{\infty }\mathop{\sum }_{k=1}^{\min (n,m-1)}a_{n-k}(Z)C_{k}^{(p^{n-k})}\blacktriangle ^{n}.\end{eqnarray}$$

We can write $v_{Z}$ in the form $v_{Z}=\sum _{n=0}^{\infty }\unicode[STIX]{x1D708}_{n}\blacktriangle ^{n}$ with $\unicode[STIX]{x1D708}_{n}\in \text{M}_{d}({\mathcal{O}})$ , $\unicode[STIX]{x1D708}_{0}=I$ . Since $\widehat{u}=pI-I\blacktriangle ^{m}$ , one gets $I\blacktriangle ^{km}\equiv p^{k}I\;\text{mod}\,\text{M}_{d}(E)\widehat{u}$ and therefore $v_{Z}\equiv \sum _{i=0}^{m-1}\unicode[STIX]{x1D6FC}_{i}\blacktriangle ^{i}\;\text{mod}\,\text{M}_{d}(E)\widehat{u}$ where $\unicode[STIX]{x1D6FC}_{i}=\sum _{j=0}^{\infty }p^{j}\unicode[STIX]{x1D708}_{jm+i}\in \text{M}_{d}({\mathcal{O}})$ . This means that there exists $r\in \text{M}_{d}(E)$ such that $\sum _{i=0}^{m-1}\unicode[STIX]{x1D6FC}_{i}\blacktriangle ^{i}=v_{Z}+r\widehat{u}=v_{Z}(I+v_{Z}^{-1}r\widehat{u})$ . Thus $\sum _{i=0}^{m-1}\unicode[STIX]{x1D6FC}_{i}\blacktriangle ^{i}\in v_{Z}(I+\text{M}_{d}(E)\widehat{u})$ , and it is obviously a unique element of this form in $v_{Z}(I+\text{M}_{d}(E)\widehat{u})$ . Moreover, $\unicode[STIX]{x1D6FC}_{0}\equiv I\hspace{0.6em}{\rm mod}\hspace{0.2em}p$ which implies that $\unicode[STIX]{x1D6FC}_{0}$ is invertible in $\text{M}_{d}({\mathcal{O}})$ and $\unicode[STIX]{x1D6FC}_{0}^{-1}\equiv I\hspace{0.6em}{\rm mod}\hspace{0.2em}p$ . For $1\leqslant i\leqslant m-1$ , denote $\unicode[STIX]{x1D6FD}_{i}=\unicode[STIX]{x1D6FC}_{0}^{-1}\unicode[STIX]{x1D6FC}_{i}$ . Then

$$\begin{eqnarray}I+\mathop{\sum }_{i=1}^{m-1}\unicode[STIX]{x1D6FD}_{i}\blacktriangle ^{i}=\unicode[STIX]{x1D6FC}_{0}^{-1}\mathop{\sum }_{i=0}^{m-1}\unicode[STIX]{x1D6FC}_{i}\blacktriangle ^{i}\in (I+\text{M}_{d}(p{\mathcal{O}}))v_{Z}(I+\text{M}_{d}(E)\widehat{u}).\end{eqnarray}$$

We consider $\unicode[STIX]{x1D6FD}_{i}$ , $1\leqslant i\leqslant m-1$ , as functions of $(\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D713}})_{\unicode[STIX]{x1D713}\in \unicode[STIX]{x1D6F9}^{+}}$ and establish explicit formulas for them.

Proposition 4.5. For any $(\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D713}})_{\unicode[STIX]{x1D713}\in \unicode[STIX]{x1D6F9}^{+}}$ , $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D713}}\in p{\mathcal{O}}$ , $1\leqslant i\leqslant m-1$

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{i}=\lim _{k\rightarrow \infty }p^{k}a_{km+i}(Z),\qquad \unicode[STIX]{x1D6FD}_{i}=\lim _{k\rightarrow \infty }a_{km}^{-1}(Z)a_{km+i}(Z).\end{eqnarray}$$

Proof. We write as above $v_{Z}=\sum _{n=0}^{\infty }\unicode[STIX]{x1D708}_{n}\blacktriangle ^{n}$ , $\unicode[STIX]{x1D708}_{n}\in \text{M}_{d}({\mathcal{O}})$ . Then $\unicode[STIX]{x1D708}_{0}=I$ and

$$\begin{eqnarray}\unicode[STIX]{x1D708}_{n}=p^{-1}\mathop{\sum }_{k=1}^{\min (n,m-1)}a_{n-k}(Z)C_{k}^{(p^{n-k})}.\end{eqnarray}$$

According to Lemma 3.1, $\unicode[STIX]{x1D708}_{n}=a_{n}(Z)$ for $0\leqslant n\leqslant m-1$ , $\unicode[STIX]{x1D708}_{n}=a_{n}(Z)-p^{-1}a_{n-m}(Z)$ for $n\geqslant m$ . These formulas imply

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}_{i} & = & \displaystyle \mathop{\sum }_{j=0}^{\infty }p^{j}\unicode[STIX]{x1D708}_{jm+i}=\lim _{k\rightarrow \infty }\mathop{\sum }_{j=0}^{k}p^{j}\unicode[STIX]{x1D708}_{jm+i}\nonumber\\ \displaystyle & = & \displaystyle \lim _{k\rightarrow \infty }\left(a_{i}(Z)+\mathop{\sum }_{j=1}^{k}p^{j}(a_{jm+i}(Z)-p^{-1}a_{(j-1)m+i}(Z))\right)\nonumber\\ \displaystyle & = & \displaystyle \lim _{k\rightarrow \infty }p^{k}a_{km+i}(Z),\nonumber\end{eqnarray}$$

for any $0\leqslant i\leqslant m-1$ . The required formula for $\unicode[STIX]{x1D6FD}_{i}$ follows immediately.◻

Remark that for any $0\leqslant i\leqslant m-1$ , the infinite sequence $p^{k}a_{km+i}(\widehat{\unicode[STIX]{x1D6EF}}^{\ast })\in \text{M}_{d}({\mathcal{O}}[[t_{\unicode[STIX]{x1D713}}]]_{\unicode[STIX]{x1D713}\in \unicode[STIX]{x1D6F9}^{+}})$ of matrices of polynomials converges in the rigid metric to a matrix of rigid analytic functions. Thus $\unicode[STIX]{x1D6FC}_{i}$ can be considered as a matrix of rigid analytic functions of variables $(\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D713}})_{\unicode[STIX]{x1D713}\in \unicode[STIX]{x1D6F9}^{+}}$ .

When $d=1$ , the functions $\unicode[STIX]{x1D6FC}_{i}$ , $0\leqslant i\leqslant m-1$ , coincide up to a constant factor with the homogeneous coordinates of the $p$ -adic period map introduced by Gross and Hopkins; see [Reference Gross and Hopkins8, 21.6, 21.13 and 23.6].

5 Action of the automorphism group

Let $F$ be a deformation of $\unicode[STIX]{x1D6F7}$ over ${\mathcal{O}}$ . For $\unicode[STIX]{x1D719}\in \text{Aut}_{k}(\unicode[STIX]{x1D6F7})$ , we put $[F]^{\prime }\unicode[STIX]{x1D719}=[g^{-1}(F(g,g))]^{\prime }$ , where $g\in {\mathcal{O}}[[X]]_{0}^{d}$ is such that its reduction is equal to $\unicode[STIX]{x1D719}$ . This defines a right action of $\text{Aut}_{k}(\unicode[STIX]{x1D6F7})$ on the set of $\star$ -isomorphism classes of the deformations of $\unicode[STIX]{x1D6F7}$ over ${\mathcal{O}}$ . In the case of dimension one, this definition is due to Lubin and Tate [Reference Lubin and Tate12]. We give an explicit description of this action in terms of the parametrization provided by Theorem 3.2(2).

It is easy to notice that the correspondence $(I+\text{M}_{d}(p{\mathcal{O}}))v(I+\text{M}_{d}(E)\widehat{u})\mapsto \text{GL}_{d}({\mathcal{O}})v(I+\text{M}_{d}(E)\widehat{u})$ is a bijection from $I+\text{M}_{d}(p{\mathcal{O}})\backslash I+\text{M}_{d}(p{\mathcal{O}}+\blacktriangle E)/I+\text{M}_{d}(E)\widehat{u}$ to $\text{GL}_{d}({\mathcal{O}})\backslash \text{GL}_{d}(E)/I+\text{M}_{d}(E)\widehat{u}$ . Thus we can identify these two sets. Define a right action of $\text{Aut}_{k}(\unicode[STIX]{x1D6F7})$ on $\text{GL}_{d}({\mathcal{O}})\backslash \text{GL}_{d}(E)/I+\text{M}_{d}(E)\widehat{u}$ in the following way: by Theorem 2.5(3), for any $\unicode[STIX]{x1D719}\in \text{Aut}_{k}(\unicode[STIX]{x1D6F7})$ there exists $w\in \text{GL}_{d}(E)$ such that $\unicode[STIX]{x1D719}$ is equal to the reduction of $\widehat{f}^{-1}(w\widehat{f})$ modulo $p$ . For $v\in \text{GL}_{d}(E)$ , put $\text{GL}_{d}({\mathcal{O}})v(I+\text{M}_{d}(E)\widehat{u})\unicode[STIX]{x1D719}=\text{GL}_{d}({\mathcal{O}})vw(I+\text{M}_{d}(E)\widehat{u})$ . Theorem 2.5(1) implies that this expression does not depend on the choice of $v$ in the corresponding coset. If $w^{\prime }\in \text{GL}_{d}(E)$ is such that the reduction of $\widehat{f}^{-1}(w^{\prime }\widehat{f})$ modulo $p$ is equal to $\unicode[STIX]{x1D719}$ , then by Lemma 2.1(1), $w\widehat{f}\equiv w^{\prime }\widehat{f}\hspace{0.6em}{\rm mod}\hspace{0.2em}\mathfrak{P}$ and Proposition 2.4 implies that there exists $s\in \text{M}_{d}(E)$ such that $w^{\prime }-w=s\widehat{u}$ , i.e., $w^{\prime }=w(1+w^{-1}s\widehat{u})\in w(I+\text{M}_{d}(E)\widehat{u})$ . Thus the definition does not depend on the choice of $w$ .

Proof. According to Theorem 5.1, Proposition 4.4 and the definition of the action of $\unicode[STIX]{x1D719}\in \text{Aut}_{k}(\unicode[STIX]{x1D6F7})$ on $\text{GL}_{d}({\mathcal{O}})\backslash \text{GL}_{d}(E)/I+\text{M}_{d}(E)\widehat{u}$ , we get

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}^{\prime }([F_{V(Z^{(2)})}]^{\prime }) & = & \displaystyle \unicode[STIX]{x1D712}^{\prime }([F_{V(Z^{(1)})}]^{\prime }\unicode[STIX]{x1D719})=\unicode[STIX]{x1D712}^{\prime }([F_{V(Z^{(1)})}]^{\prime })\unicode[STIX]{x1D719}\nonumber\\ \displaystyle & = & \displaystyle \text{GL}_{d}({\mathcal{O}})v_{Z^{(1)}}(I+\,\text{M}_{d}(E)\widehat{u})\unicode[STIX]{x1D719}=\text{GL}_{d}({\mathcal{O}})v_{Z^{(1)}}w(I+\,\text{M}_{d}(E)\widehat{u}).\nonumber\end{eqnarray}$$

On the other hand, $\unicode[STIX]{x1D712}^{\prime }([F_{V(Z^{(2)})}]^{\prime })=\text{GL}_{d}({\mathcal{O}})v_{Z^{(2)}}(I+\text{M}_{d}(E)\widehat{u})$ . This implies the required statement.◻

Corollary.

In the setting of Theorem 5.2, suppose that $\unicode[STIX]{x1D6EF}_{m}=I$ , $\unicode[STIX]{x1D6EF}_{n}=0$ for $n\neq m$ , and choose $w_{i}\in \text{M}_{d}({\mathcal{O}})$ such that $w\equiv \sum _{i=0}^{m-1}w_{i}\blacktriangle ^{i}\;\text{mod}\,\text{M}_{d}(E)\widehat{u}$ . Then $[F_{V(Z^{(1)})}]^{\prime }\unicode[STIX]{x1D719}=[F_{V(Z^{(2)})}]^{\prime }$ iff there exists $c\in \text{GL}_{d}({\mathcal{O}})$ such that

$$\begin{eqnarray}\left(\unicode[STIX]{x1D6FC}_{0}^{(2)},\ldots ,\unicode[STIX]{x1D6FC}_{m-1}^{(2)}\right)=c\left(\unicode[STIX]{x1D6FC}_{0}^{(1)},\ldots ,\unicode[STIX]{x1D6FC}_{m-1}^{(1)}\right)C(w),\end{eqnarray}$$

where

$$\begin{eqnarray}C(w)=\left(\begin{array}{@{}ccccc@{}}w_{0} & w_{1} & w_{2} & & w_{m-1}\\ pw_{m-1}^{\unicode[STIX]{x1D6E5}} & w_{0}^{\unicode[STIX]{x1D6E5}} & w_{1}^{\unicode[STIX]{x1D6E5}} & \cdots \, & w_{m-2}^{\unicode[STIX]{x1D6E5}}\\ pw_{m-2}^{\unicode[STIX]{x1D6E5}^{2}} & pw_{m-1}^{\unicode[STIX]{x1D6E5}^{2}} & w_{0}^{\unicode[STIX]{x1D6E5}^{2}} & & w_{m-3}^{\unicode[STIX]{x1D6E5}^{2}}\\ & \vdots & & \ddots & \vdots \\ pw_{1}^{\unicode[STIX]{x1D6E5}^{m-1}} & pw_{2}^{\unicode[STIX]{x1D6E5}^{m-1}} & pw_{3}^{\unicode[STIX]{x1D6E5}^{m-1}} & \cdots \, & w_{0}^{\unicode[STIX]{x1D6E5}^{m-1}}\end{array}\right).\end{eqnarray}$$

Proof. As we showed above $\sum _{i=0}^{m-1}\unicode[STIX]{x1D6FC}_{i}^{(2)}\blacktriangle ^{i}\in v_{Z^{(2)}}(I+\text{M}_{d}(E)\widehat{u})$ . Besides, Theorem 5.2 implies

$$\begin{eqnarray}\displaystyle \text{GL}_{d}({\mathcal{O}})v_{Z^{(2)}}(I+\text{M}_{d}(E)\widehat{u}) & = & \displaystyle \text{GL}_{d}({\mathcal{O}})v_{Z^{(1)}}w(I+\text{M}_{d}(E)\widehat{u})\nonumber\\ \displaystyle & = & \displaystyle \text{GL}_{d}({\mathcal{O}})\left(\mathop{\sum }_{i=0}^{m-1}\unicode[STIX]{x1D6FC}_{i}^{(1)}\blacktriangle ^{i}\right)w(I+\text{M}_{d}(E)\widehat{u}).\nonumber\end{eqnarray}$$

Hence, there exist $c\in \text{GL}_{d}({\mathcal{O}})$ and $q\in \text{M}_{d}(E)$ such that

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{i=0}^{m-1}\unicode[STIX]{x1D6FC}_{i}^{(2)}\blacktriangle ^{i} & = & \displaystyle c\left(\mathop{\sum }_{i=0}^{m-1}\unicode[STIX]{x1D6FC}_{i}^{(1)}\blacktriangle ^{i}\right)w(1+q\widehat{u})\nonumber\\ \displaystyle & \equiv & \displaystyle c\left(\mathop{\sum }_{i=0}^{m-1}\unicode[STIX]{x1D6FC}_{i}^{(1)}\blacktriangle ^{i}\right)w\;\text{mod}\,\text{M}_{d}(E)\widehat{u}.\nonumber\end{eqnarray}$$

It is easy to check directly that

$$\begin{eqnarray}\left(\mathop{\sum }_{i=0}^{m-1}\unicode[STIX]{x1D6FC}_{i}^{(1)}\blacktriangle ^{i}\!\right)\!w\equiv \left(\unicode[STIX]{x1D6FC}_{0}^{(1)},\ldots ,\unicode[STIX]{x1D6FC}_{m-1}^{(1)}\right)C(w)(1,\blacktriangle ,\ldots ,\blacktriangle ^{m-1})^{T}\;\text{mod}\,\text{M}_{d}(E)\widehat{u}.\end{eqnarray}$$

Therefore

$$\begin{eqnarray}\mathop{\sum }_{i=0}^{m-1}\unicode[STIX]{x1D6FC}_{i}^{(2)}\blacktriangle ^{i}\equiv c\left(\unicode[STIX]{x1D6FC}_{0}^{(1)},\ldots ,\unicode[STIX]{x1D6FC}_{m-1}^{(1)}\right)C(w)\left(1,\blacktriangle ,\ldots ,\blacktriangle ^{m-1}\right)^{T}\;\text{mod}\,\text{M}_{d}(E)\widehat{u}.\end{eqnarray}$$

This means that there exists $r\in \text{M}_{d}(E)$ such that

$$\begin{eqnarray}\mathop{\sum }_{i=0}^{m-1}\unicode[STIX]{x1D6FC}_{i}^{(2)}\blacktriangle ^{i}-c\left(\unicode[STIX]{x1D6FC}_{0}^{(1)},\ldots ,\unicode[STIX]{x1D6FC}_{m-1}^{(1)}\right)C(w)\left(1,\blacktriangle ,\ldots ,\blacktriangle ^{m-1}\right)^{T}=r\widehat{u}.\end{eqnarray}$$

Since the highest power of $\blacktriangle$ which appears in the left side of the equality is less or equal to $m-1$ , we conclude that $r$ must be equal to $0$ , which implies the required statement.◻

In the case $d=1$ , the last Corollary provides the result by Gross and Hopkins on equivariance of their $p$ -adic period map [Reference Gross and Hopkins8, Proposition 23.5].