I am trying to understand how to see whether a given formal group is $p$-divisible. Let $A$ be a complete noetherian local ring with maximal ideal $\mathfrak{m}$ and residue field $k$ of characteristic $p$ and let $F \in A[[X,Y]]$ be a formal group law.

Is there a nice criterion to see whether the map $$[p]^* \colon A[[X]] \to A[[X]], f(X) \mapsto f([p](X))$$ is injective and makes $A[[X]]$ into a finite free module over $A[[X]]$? (This is the definition of $F$ being $p$-divisible.)

My main case of interest is when $A=\mathfrak{o}$ is the ring of integers in a finite extension of $\mathbb Q_p$. Lubin's answer to the question How is the $p$-adic Tate module of a formal group defined? seems to imply that

$$F \,\,\text{ is } p\text{ -divisible } \iff [p] \text{ mod } \pi \neq 0 \text{ in } k[[X]]$$

where $\pi$ denotes a uniformizer of $\mathfrak o$. Is this true? I would very much be thankful for a proof of this!


There are probably several proofs of this, and I would almost make a wager that the slickest of these would dispose of the matter with the right appeal to Nakayama’s Lemma. But let me be historical and tell you how I have looked at it.

I’m supposing that you’re aware that we’re always dealing with formal groups which are not isomorphic to the additive formal group $x+y$ over $k$. And I’m supposing that you know the result (due to Lazard, I suppose) that the first nonzero coefficient of $[p]_{\tilde F}$ (where $\tilde F$ is the formal group over $k$) must be in degree $p^h$, and that this $h$ is called the height of $F$.

This means that the first unit coefficient of $[p]_F$ appears in degree $p^h$.

Let $f=[p]$. Now, within $A[[x]]$ we have the subring $A[[f]]$, and I think you can easily see that $A[[f]]\cong A[[T]]$, power-series ring in one indeterminate over $A$. So, to avoid confusion, I’m going to name $f(x)=T$, so that we have $A[[T]]\subset A[[x]]$, and I am about to show that the big ring is free over $A[[T]]$. Consider $A[[T]][[X]]\big/\bigl(f(X)-T\bigr)$, which I think you see is isomorphic to $A[[x]]$ via $X\mapsto x$. But now I would call in Weierstrass Preparation, in this form: Let $\mathcal O$ be a complete local ring, and and let $\phi(X)\in\mathcal O[[X]]$ have first unit coefficient in degree $n$. Then there is, uniquely, a pair $(g,U)$ where $g\in\mathcal O[X]$, monic polynomial of degree $n$; and $U\in\mathcal O[[X]]^\times$, power series with constant term a unit of $\mathcal O$, such that $\phi=gU$.

And of course that does it for us, letting $\phi=f(X)-T$ above, and $\mathcal O=A[[T]]$.

If you have further questions, don’t hesitate to ask. As for W-Prep, I claim that every mathematician that sits down to prove it will give a different proof. It is definitely not as deep a fact as Hensel’s Lemma.

  • $\begingroup$ A very nice proof, as always. So we conclude that a formal group is $p$-divisible iff it is of finite height. I am trying to get familiar with Tate's paper on $p$-divisible groups. It is a very significant, groundbreaking paper, as I understand. It would be great if you could give your opinion on why this is so, i.e. name a particularly nice result of the paper and what it implies for formal groups of finite height? $\endgroup$ – Layer Cake Mar 19 at 15:01
  • $\begingroup$ Thanks for the edits, @LayerCake. I’m a terrible proofreader. I’m not sure I’m qualified to discuss the significance of anything; I suppose one might want to look at that paper for what it may tell you about Abelian varieties. $\endgroup$ – Lubin Mar 19 at 18:00
  • 2
    $\begingroup$ @LayerCake There are several papers / results that, when taken together, were the germ of what became $p$-adic Hodge theory. Often people mention only Tate's paper as being the origin of the theory because there appears the famous Hodge-Tate decomposition for the Tate module of an Abelian variety, but it's not accurate to cite this alone. The theory really emerged out of comparing Tate's results with Grothendieck's results in Groupes de Barsotti-Tate et Cristaux. Underlying this whole thing was the search for an "$\ell =p$'' analogue of the Néron-Ogg-Shafarevich criterion. $\endgroup$ – Giovanni Di Matteo Mar 21 at 23:38
  • $\begingroup$ Thanks, @GiovanniDiMatteo, for this valuable comment. $\endgroup$ – Lubin Mar 22 at 3:45
  • $\begingroup$ Let me be a little more precise. Let $K/\mathbb{Q}_p$ be a finite extension, $k$ its residue field. N-O-S says that for $\ell\neq p$, $A/K$ (abelian variety) has good reduction iff $T_\ell(A)$ is non-ramified as a representation of $G_K=\text{Gal}(\overline{\mathbb{Q}}_p/K)$. The analogue of this statement is false for $\ell=p$ (for example, if $E$ is an elliptic curve over $\mathbb{Q}_p$ then $\det(T_p(E))$ is the cyclotomic character - i.e. very ramified, and thus so is $T_p(E)$). Serre & Tate's proof of N-O-S also breaks down when $\ell = p$, because the $p$-adic Tate module in char $p$ ... $\endgroup$ – Giovanni Di Matteo Mar 30 at 22:25

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