## Fun with crystalline periods

The earliest results in what we now call p-adic Hodge theory are due to Tate and Sen, but nevertheless it seems reasonable to largely credit Fontaine for giving shape and direction to this subject and establishing it as a vital, distinctive branch of…um… number theory? arithmetic geometry? ring theory? Like I said, distinctive. One of Fontaine’s big innovations was the introduction of a bunch of interesting period rings with Galois actions and other auxiliary structures, which lead (among other things) to meaningful classifications of p-adic Galois representations.

I don’t think I’ll be accused of criticism if I say that Fontaine’s rings are not exactly intuitively warm and inviting objects when you first meet them; their actual definitions seem like a crazy jumble of operations applied arbitrarily to a sequence of wilder and wilder rings. I actually remember exactly where and when I first heard the definition of ${\mathbf{B}_{\mathrm{dR}}}$: at a Thai restaurant in Brookline with JT, in the spring of 2011. The conversation probably went something like this (my thoughts are in italics):

D. So what is ${\mathbf{B}_{\mathrm{dR}}}$, anyway?

J. Well, you start with the ring of integers of an algebraic closure of ${\mathbf{Q}_{p}}$.

D. Okay.

J. Then you reduce this modulo p, and take the inverse limit of this along the Frobenius action.

D. (Huh?)

J. Then you take the Witt vectors of this and invert p.

D. (?!?)

J. It turns out this ring has a natural surjection onto ${\mathbf{C}_{p}}$, and the kernel is a principal ideal. The completion for the topology defined by powers of this ideal is ${\mathbf{B}_{\mathrm{dR}}^{+}}$.

D. (What have I gotten myself into? When will it end?!?!)

J. Then you invert any generator of this ideal, and that’s ${\mathbf{B}_{\mathrm{dR}}}$.

D. Hmm, interesting. (I wonder if there are any monasteries around here I could join. Or maybe it’s not too late to switch to zookeeping?)

All joking aside, I think these sorts of thoughts are pretty common, and understandable; there’s plenty of objects out there which seem so complicated on first exposure that it’s hard to believe any human came up with them, let alone that you can do meaningful calculations with them. But then you slowly get used to them with repeated exposure, and one day you find yourself merrily calculating away. Which brings me to the goal of this post: an explicit, motivated step-by-step calculation with one of the more delicate Fontaine rings, namely ${\mathbf{A}_{\mathrm{crys}}}$. (The actual definition is below.)

Okay, so let ${K}$ be a finite, unramified extension of ${\mathbf{Q}_{p}}$ with degree ${d}$ and ring of integers ${\mathcal{O}}$; fix once and for all an algebraic closure ${\overline{K}}$ with completion ${\mathbf{C}=\widehat{\overline{K}}}$. Let ${\sigma}$ be the unique automorphism of ${K}$ which lifts the Frobenius on ${\mathcal{O}/p}$. Let ${G=\mathrm{Spf}\mathcal{O}[[T]]}$ be a p-divisible commutative formal group of dimension one and height ${h}$ over ${\mathcal{O}}$; explicitly, we fix a formal group law ${X+_{G}Y\in\mathcal{O}[[X,Y]]}$ such that, writing ${[m](X)}$ for the ${m}$-fold sum ${X+_{G}+\dots+_{G}X}$, ${T\mapsto[p](T)}$ induces an isogeny of degree ${p^{h}}$. As usual, ${G}$ defines a functor from p-adically complete ${\mathcal{O}}$-algebras to abelian groups, and in particular ${G(\mathcal{O}_{\mathbf{C}})=(\mathfrak{m}_{\mathbf{C}},+_{G})}$. Let us set

$\displaystyle \begin{array}{rcl} \tilde{G}(\mathcal{O}_{\mathbf{C}}) & = & \lim_{\leftarrow\cdot p}G(\mathcal{O}_{C})\\ & = & \left\{ x=(x_{0},x_{1},x_{2},\dots)\mid x_{i}\in\mathfrak{m}_{\mathbf{C}}\,\mathrm{and}\,[p](x_{i+1})=x_{i}\right\} ,\end{array}$

so we have a natural short exact sequence

$\displaystyle 0\rightarrow T_{p}G\rightarrow\tilde{G}(\mathcal{O}_{\mathbf{C}})\overset{x\mapsto x_{0}}{\rightarrow}G(\mathcal{O}_{\mathbf{C}})\rightarrow0.$

Inside ${G}$ we have the subgroup defined by ${G^{+}(\mathcal{O}_{\mathbf{C}})=\left\{ x\in G(\mathcal{O}_{\mathbf{C}})\mid v(x_{0})\geq1\right\} }$. Set ${\tilde{G}^{+}(\mathcal{O}_{C})=\{x\in\tilde{G}(\mathcal{O}_{\mathbf{C}})\mid x_{0}\in G^{+}(\mathcal{O}_{C})\}}$.

Let ${\omega\in\mathcal{O}[[X]]dX}$ be a differential of the second kind for ${G}$, so by definition ${\omega=dF_{\omega}}$ where ${F_{\omega}(X)\in XK[[X]]}$ is a power series with ${F_{\omega}(X+_{G}Y)-F_{\omega}(X)-F_{\omega}(Y)\in\mathcal{O}[[X,Y]]}$. A differential is invariant if ${F_{\omega}(X+_{G}Y)-F_{\omega}(X)-F_{\omega}(Y)=0}$. Two differentials of the second kind are equivalent if ${F_{\omega_{1}}-F_{\omega_{2}}\in X\mathcal{O}[[X]]}$, and the Dieudonne module ${\mathbf{D}(G)}$ is defined as the ${\mathcal{O}}$-module of equivalence classes of differentials. This turns out to be a free ${\mathcal{O}}$-module of rank ${h}$ with some interesting structures on it: for example, you can check by hand that the formula ${F_{\varphi(\omega)}(X)=F_{\omega}^{\sigma}(X^{p})}$ defines an ${\mathcal{O}}$-semilinear Frobenius action on ${\mathbf{D}(G)}$. Fancier types will recognize ${\mathbf{D}(G)}$ more naturally as the cotangent space of the universal vector extension of ${G}$. In particular, ${\mathbf{D}(G)}$ sits in a short exact sequence

$\displaystyle 0\rightarrow\omega_{G}\rightarrow\mathbf{D}(G)\rightarrow\mathrm{Lie}(G^{\ast})\rightarrow0,$

where ${\omega_{G}}$ is the line of invariant differentials and ${G^{\ast}}$ is the dual p-divisible group.

Now some rings. I’ll give the raw definitions quickly, since this is done slowly in many places in the literature (cf. especially some great survey papers by Berger). Set

$\displaystyle \tilde{\mathbf{E}}^{+}=\mathcal{O}_{\mathbf{C}}^{\flat}=\left\{ (a^{(0)},a^{(1)},\dots)\mid a^{(i)}\in\mathcal{O}_{\mathbf{C}}\,\mathrm{with}\,(a^{(i+1)})^{p}=a^{(i)}\right\} .$

Reduction modulo ${p}$ identifies this isomorphically as a multiplicative monoid with the inverse limit of ${\mathcal{O}_{\mathbf{C}}/p}$ along the Frobenius, and the latter has an obvious ring structure which transports back to a ring structure on ${\tilde{\mathbf{E}}^{+}}$. Set ${\tilde{\mathbf{A}}^{+}=W(\tilde{\mathbf{E}}^{+})}$ and ${\tilde{\mathbf{B}}^{+}=\tilde{\mathbf{A}}^{+}[\frac{1}{p}]}$. The ring ${\tilde{\mathbf{A}}^{+}}$ is p-adically complete and p-torsion free, and the natural homomorphism

$\displaystyle \begin{array}{rcl} \theta:\tilde{\mathbf{E}}^{+} & \rightarrow & \mathcal{O}_{\mathbf{C}}/p\\ a & \mapsto & \overline{a^{(0)}}\end{array}$

lifts to a surjective ring homomorphism

$\displaystyle \begin{array}{rcl} \theta:\tilde{\mathbf{A}}^{+} & \rightarrow & \mathcal{O}_{\mathbf{C}}\\ \sum_{i\geq0}p^{i}[a_{i}] & \mapsto & \sum_{i\geq0}p^{i}a_{i}^{(0)}.\end{array}$

It turns out the kernel of ${\theta}$ is principal, so let’s fix a generator ${\xi}$. The ring ${\tilde{\mathbf{A}}^{+}}$ is complete for the ${\xi}$-adic topology, and thus for the ${(\xi,p)}$-adic topology. The ring ${\mathbf{A}_{\mathrm{crys}}^{0}}$ is the divided power envelope of ${\tilde{\mathbf{A}}^{+}}$ with respect to the ideal ${(\xi)}$, so ${\mathbf{A}_{\mathrm{crys}}^{0}=\tilde{\mathbf{A}}^{+}[\dots,\frac{\xi^{i}}{i!},\dots]}$. You can think of ${\mathbf{A}_{\mathrm{crys}}^{0}}$ as a subring of ${\tilde{\mathbf{B}}^{+}}$ if you like. Finally, ${\mathbf{A}_{\mathrm{crys}}}$ is the p-adic completion of ${\mathbf{A}_{\mathrm{crys}}^{0},}$ and ${\mathbf{B}_{\mathrm{crys}}^{+}=\mathbf{A}_{\mathrm{crys}}[\frac{1}{p}]}$. It turns out that ${\mathbf{A}_{\mathrm{crys}}}$ is p-torsion free (not so obvious!), so ${\mathbf{A}_{\mathrm{crys}}\subset\mathbf{B}_{\mathrm{crys}}^{+}}$. The map ${\theta}$ still exists out of these rings, and the Frobenius on ${\tilde{\mathbf{A}}^{+}}$ extends to them.

Given any ${x=(x_{i})_{i\geq0}}$ in ${\tilde{G}^{+}(\mathcal{O}_{\mathbf{C}})}$, let ${\tilde{x}_{i}\in\tilde{\mathbf{A}}^{+}}$ denote a choice of any element lifting ${x_{i}}$ under the map ${\theta}$.

Theorem. For any ${x\in\tilde{G}^{+}(\mathcal{O}_{\mathbf{C}})}$ and any ${\omega}$, the limit

$\displaystyle \rho_{x}(\omega)=\mathrm{lim}_{n\rightarrow\infty}p^{n}F_{\omega}(\tilde{x}_{n})$

exists in ${\mathbf{A}_{\mathrm{crys}}}$ and is independent of the choices of lifts, and depends only on the equivalence class of ${\omega}$. This limit therefore defines a natural pairing

$\displaystyle \begin{array}{rcl} \tilde{G}^{+}(\mathcal{O}_{\mathbf{C}})\otimes\mathbf{D}(G) & \rightarrow & \mathbf{A}_{\mathrm{crys}}\\ x\otimes\omega & \mapsto & \rho_{x}(\omega)\end{array}$

which furthermore is nondegenerate and which satisfies ${\rho_{x}(\varphi(\omega))=\varphi(\rho_{x}(\omega))}$. If ${x\in T_{p}G}$ and ${\omega\in\omega_{G}}$ then ${\theta(\rho_{x}(\omega))=0}$.

Looking at the proof below, it’s immediate that this is the restriction of a more general pairing ${\tilde{G}(\mathcal{O}_{\mathbf{C}})\otimes\mathbf{D}(G)\rightarrow\mathbf{B}_{\mathrm{crys}}^{+}}$, but I’ll stick with the integral formulation above. In any case, there are several nice ways to hold this up to the light. First of all, we get an injection

$\displaystyle \begin{array}{rcl} T_{p}G & \hookrightarrow & \mathrm{Hom}_{\varphi,\mathrm{Fil}}(\mathbf{D}(G),\mathbf{A}_{\mathrm{crys}})\\ x & \mapsto & \rho_{x}(\cdot),\end{array}$

where the subscript indicates ${\varphi}$-equivariant ${\mathbf{Z}_{p}}$-linear maps which carry ${\omega_{G}}$ into ${\ker\theta}$. This turns out to be an isomorphism of Galois modules (at least if ${p>2}$ or something), and is maybe the simplest incarnation of Fontaine-Laffaille theory. Second of all, since ${\mathrm{Hom}_{\mathrm{Fil}}(\mathbf{D}(G),\mathbf{A}_{\mathrm{crys}})}$ is by definition the kernel of the map

$\displaystyle \begin{array}{rcl} \theta_{G}:\mathrm{Hom}(\mathbf{D}(G),\mathbf{A}_{\mathrm{crys}}) & \rightarrow & \mathrm{Lie}(G)\otimes\mathcal{O}_{\mathbf{C}}\\ f & \mapsto & \theta\circ f|_{\omega_{G}},\end{array}$

we get a complex

$\displaystyle 0\rightarrow T_{p}G\rightarrow\mathrm{Hom}_{\varphi}(\mathbf{D}(G),\mathbf{A}_{\mathrm{crys}})\overset{\theta_{G}}{\rightarrow}\mathrm{Lie}(G)\otimes\mathcal{O}_{\mathbf{C}}\rightarrow0.$

Taking Galois invariants gives a connecting homomorphism

$\displaystyle \delta_{G}:(\mathrm{im}\theta_{G})^{G_{K}}\rightarrow H^{1}(K,T_{p}G)$

in Galois cohomology, a primordial Bloch-Kato exponential. On the other hand, we have the sequence

$\displaystyle 0\rightarrow G[p^{n}](\mathcal{O}_{\mathbf{C}})\rightarrow G(\mathcal{O}_{\mathbf{C}})\overset{\cdot p^{n}}{\rightarrow}G(\mathcal{O}_{\mathbf{C}})\rightarrow0$

of ${G_{K}}$-modules. Passing to ${G_{K}}$-invariants and taking the inverse limit over ${n}$ induces the more familiar Kummer connecting homomorphism

$\displaystyle \kappa_{G}:G(\mathcal{O}_{K})\rightarrow H^{1}(K,T_{p}G)$

in Galois cohomology, and it turns out that ${\kappa_{G}=\delta_{G}\circ\log_{G}}$. In fact this is pretty immediate from the diagram

$\displaystyle \begin{array}{rcl} 0\rightarrow T_{p}G & \longrightarrow\tilde{G}^{+}(\mathcal{O}_{\mathbf{C}})\longrightarrow & G^{+}(\mathcal{O}_{C})\rightarrow0\\ \parallel\; & \downarrow & \;\downarrow\\ 0\rightarrow T_{p}G & \rightarrow\mathrm{Hom}_{\varphi}(\mathbf{D}(G),\mathbf{A}_{\mathrm{crys}})\rightarrow & \mathrm{Lie}(G)\otimes\mathcal{O}_{\mathbf{C}}\rightarrow0\end{array}$

where the middle downwards arrow sends ${x}$ to ${\rho_{x}}$ and the rightmost downwards arrow is ${\log_{G}}$. (Sorry for the terrible formatting!) This comparison vividly illustrates why ${\kappa_{G}}$ lands in ${H_{f}^{1}}$.

Now the proof. It turns out that we can give a fairly soft proof by dividing it into a few steps.

Step One. The series ${F_{\omega}(\tilde{x}_{n})}$ converges to a well-defined element of ${\mathbf{B}_{\mathrm{crys}}^{+}}$. Furthermore, ${F_{\omega}(\tilde{x}_{0})\in\mathbf{A}_{\mathrm{crys}}}$.

Step Two. For any ${n\geq0}$, ${p^{n+1}F_{\omega}(\tilde{x}_{n+1})-p^{n}F_{\omega}(\tilde{x}_{n})\in p^{n}\mathbf{A}_{\mathrm{crys}}}$.

Step Three. The image of ${p^{n}F_{\omega}(\tilde{x}_{n})}$ in ${\mathbf{A}_{\mathrm{crys}}/p^{n}}$ is independent of the choice of ${\tilde{x}_{n}}$, and depends only on the equivalence class of ${\omega}$.

The result is immediate from here: the first two steps show ${p^{n}F_{\omega}(\tilde{x}_{n})}$ is a Cauchy sequence in the ${p}$-adic topology on ${\mathbf{A}_{\mathrm{crys}}}$, and the third step shows that its limit has the claimed properties.

Proof of Step One. In fact, something more general is true: if ${\sum_{i\geq0}a_{i}X^{i}\in K[[X]]}$ converges on the open unit disk, and ${y\in\tilde{\mathbf{A}}^{+}}$ is any element with ${v(\theta(y))>0}$, then ${\sum_{i\geq0}a_{i}y^{i}}$ converges in ${\mathbf{B}_{\mathrm{crys}}^{+}}$. To see this, note that the topology on ${\mathbf{B}_{\mathrm{crys}}^{+}}$ is defined by a natural p-adic valuation ${v_{\mathrm{crys}}}$, with ${v_{\mathrm{crys}}(y)}$ equal to the greatest integer ${n}$ such that ${y\in p^{n}\mathbf{A}_{\mathrm{crys}}}$. Under our hypotheses, we may choose some large ${M}$ with ${v(\theta(y^{M}))\geq1}$, which is to say with ${y^{M}\in(\xi,p)}$. The latter ideal certainly has divided powers in ${\mathbf{A}_{\mathrm{crys}}}$, so ${y^{Mn}\in n!\mathbf{A}_{\mathrm{crys}}}$, which implies in particular that ${v_{\mathrm{crys}}(y^{n})\gg Cn}$ for some constant ${C>0}$. Since by hypothesis ${v(a_{i})+Ci\rightarrow\infty}$ for any ${C>0}$, the claimed convergence follows.

For the more delicate second statement, note that by our assumptions we have ${\tilde{x}_{0}\in(\xi,p)}$ and so ${\tilde{x}_{0}}$ has divided powers in ${\mathbf{A}_{\mathrm{crys}}}$. Writing

$\displaystyle \begin{array}{rcl} F_{\omega}(\tilde{x}_{0}) & = & \sum_{i\geq0}a_{i}\tilde{x}_{0}^{i}\\ & = & \sum_{i\geq0}i!a_{i}\frac{\tilde{x}_{0}^{i}}{i!},\end{array}$

the claim then follows from the fact that ${ia_{i}\in\mathcal{O}_{K}}$.

Proof of Step Two. Since ${\omega}$ is a differential of the second kind, we have

$\displaystyle F_{\omega}(X+_{G}X)-2F_{\omega}(X)\in\mathcal{O}[[X]],$

so an easy induction gives

$\displaystyle pF_{\omega}(X)=F_{\omega}([p](X))+g(X)$

for some ${g\in\mathcal{O}[[X]]}$. In a similar vein, we have

$\displaystyle F_{\omega}(X)-F_{\omega}(Y)=F_{\omega}(X-_{G}Y)+h(X,Y)$

for some ${h(X,Y)\in\mathcal{O}[[X,Y]]}$. Using these identities, we rewrite the difference in question as follows:

$\displaystyle \begin{array}{rcl} p^{n+1}F_{\omega}(\tilde{x}_{n+1})-p^{n}F_{\omega}(\tilde{x}_{n}) & = & p^{n}\left(pF_{\omega}(\tilde{x}_{n+1})-F_{\omega}(\tilde{x}_{n})\right)\\ & = & p^{n}\left(g(\tilde{x}_{n+1})+F_{\omega}([p](\tilde{x}_{n+1}))-F_{\omega}(\tilde{x}_{n})\right)\\ & = & p^{n}\left(g(\tilde{x}_{n+1})+F_{\omega}([p](\tilde{x}_{n+1})-_{G}\tilde{x}_{n})+h([p](\tilde{x}_{n+1}),\tilde{x}_{n})\right).\end{array}$

It’s easy to see the ${g}$– and ${h}$-terms here converge in ${\widetilde{\mathbf{A}}^{+}}$, so all that remains is to show ${F_{\omega}([p](\tilde{x}_{n+1})-_{G}\tilde{x}_{n})\in\mathbf{A}_{\mathrm{crys}}}$. Now something great happens. Writing ${\xi_{n}=[p](\tilde{x}_{n+1})-_{G}\tilde{x}_{n}\in\widetilde{\mathbf{A}}^{+}}$, we have

$\displaystyle \begin{array}{rcl} \theta(\xi_{n}) & = & [p](x_{n+1})-_{G}x_{n}\\ & = & x_{n}-_{G}x_{n}\\ & = & 0,\end{array}$

so ${\xi_{n}\in\ker\theta}$! Therefore ${\xi_{n}}$ has divided powers in ${\mathbf{A}_{\mathrm{crys}}}$ and ${F_{\omega}(\xi_{n})=\sum i!a_{i}\frac{\xi_{n}^{i}}{i!}}$ defines an element of ${\mathbf{A}_{\mathrm{crys}}}$ exactly as step one.

Proof of Step Three.If ${\tilde{x}_{n}}$ and ${\tilde{x}_{n}'}$ are two different lifts, then

$\displaystyle F_{\omega}(\tilde{x}_{n})-F_{\omega}(\tilde{x}_{n}')=F_{\omega}(\tilde{x}_{n}-_{G}\tilde{x}_{n}')+h(\tilde{x}_{n},\tilde{x}_{n}')$

(with the notation as in step two). Since ${\tilde{x}_{n}-_{G}\tilde{x}_{n}'\in\ker\theta}$, the right-hand side of this expression gives a well-defined element of ${\mathbf{A}_{\mathrm{crys}}}$, so the independence-of-the-lift claim follows. Likewise, if ${\omega_{1}\sim\omega_{2}}$, then

$\displaystyle F_{\omega_{1}}(\tilde{x}_{n})-F_{\omega_{2}}(\tilde{x}_{n})=g(\tilde{x}_{n})\in\tilde{\mathbf{A}}^{+}$

for some ${g\in\mathcal{O}[[X]]}$, and we’re done.