# Katz's paper on Serre Tate local moduli

In katz's paper "Serre-Tate local moduli" chaper 3 has the following construction: Let $A$ be a fixed ordinary elliptic curve defined over $k$ of char $p>0$. Consider the deformation of $A$ to $W(k)$-witt vector ring. Denote by $\mathcal{M}_A$ the corresponding formal moduli space, and let $\mathcal{A}/\mathcal{M}_A$ denote the universal formal deformation of $A/k$. Let $R$ be any artin local ring with risidue field $k$, and any lifting $\mathbb{A}/R$ of $A/k$. Let $\mathbb{A}'/R$ be the quotient of $\mathbb{A}$ by the "canonical subgroup" $\hat{\mathbb{A}}[p]$ of $\mathbb{A}$. Here $\hat{\mathbb{A}}$ denote the connected part of the $p$-divisible group associated to $\mathbb{A}$.

If we apply the construction $$\mathbb{A}/R\to \mathbb{A}'/R$$ to the universal formal deformation $\mathcal{A}/\mathcal{M}_A$, we obtain a formal deformation $\mathcal{A}'/\mathcal{M}_{A^{(\sigma)}}$ of $A^{(\sigma)}/k$, here $\sigma$ is Frobenius.

I have some question about this $\mathcal{A}'/\mathcal{M}_{A^{(\sigma)}}$. I guess that it has the following property: Let $W_n(k)$ denote the witt vector ring of length n. And let $j_n$ denote the natural closed iimmersion $spec(W_{n-1}(k))\to spec(W_n(k))$. suppose we have two $W_n(k)$ point of $\mathcal{M}_{A^{(\sigma)}}$: for $i=1,2$

$s_i:spec(W_n(k))\to\mathcal{M}_{A^{(\sigma)}}$.

so we obtain two Elliptic curves $E_i$ over $W_n(k)$ .

Then we have the follwoing statement:

If $s_1\circ j_n=s_2\circ j_n$. Then $E_1$ and $E_2$ should be isomorphic over $W_n(k)$.

is this statement true? Thank you!

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I don't understand the statement $j_n\circ s_1=j_n\circ s_2$. The source of $j_n$ is not in the target of $s_i$. –  Damian Rössler Jul 24 '13 at 10:40
I'm sorry, I make a wrong order, I've corrected it. –  Lan Jul 24 '13 at 11:27