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 : 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 , anyway?
J. Well, you start with the ring of integers of an algebraic closure of .
J. Then you reduce this modulo p, and take the inverse limit of this along the Frobenius action.
J. Then you take the Witt vectors of this and invert p.
J. It turns out this ring has a natural surjection onto , and the kernel is a principal ideal. The completion for the topology defined by powers of this ideal is .
D. (What have I gotten myself into? When will it end?!?!)
J. Then you invert any generator of this ideal, and that’s .
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 . (The actual definition is below.)
Okay, so let be a finite, unramified extension of with degree and ring of integers ; fix once and for all an algebraic closure with completion . Let be the unique automorphism of which lifts the Frobenius on . Let be a p-divisible commutative formal group of dimension one and height over ; explicitly, we fix a formal group law such that, writing for the -fold sum , induces an isogeny of degree . As usual, defines a functor from p-adically complete -algebras to abelian groups, and in particular . Let us set
so we have a natural short exact sequence
Inside we have the subgroup defined by . Set .
Let be a differential of the second kind for , so by definition where is a power series with . A differential is invariant if . Two differentials of the second kind are equivalent if , and the Dieudonne module is defined as the -module of equivalence classes of differentials. This turns out to be a free -module of rank with some interesting structures on it: for example, you can check by hand that the formula defines an -semilinear Frobenius action on . Fancier types will recognize more naturally as the cotangent space of the universal vector extension of . In particular, sits in a short exact sequence
where is the line of invariant differentials and 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
Reduction modulo identifies this isomorphically as a multiplicative monoid with the inverse limit of along the Frobenius, and the latter has an obvious ring structure which transports back to a ring structure on . Set and . The ring is p-adically complete and p-torsion free, and the natural homomorphism
lifts to a surjective ring homomorphism
It turns out the kernel of is principal, so let’s fix a generator . The ring is complete for the -adic topology, and thus for the -adic topology. The ring is the divided power envelope of with respect to the ideal , so . You can think of as a subring of if you like. Finally, is the p-adic completion of and . It turns out that is p-torsion free (not so obvious!), so . The map still exists out of these rings, and the Frobenius on extends to them.
Given any in , let denote a choice of any element lifting under the map .
Theorem. For any and any , the limit
exists in and is independent of the choices of lifts, and depends only on the equivalence class of . This limit therefore defines a natural pairing
which furthermore is nondegenerate and which satisfies . If and then .
Looking at the proof below, it’s immediate that this is the restriction of a more general pairing , 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
where the subscript indicates -equivariant -linear maps which carry into . This turns out to be an isomorphism of Galois modules (at least if or something), and is maybe the simplest incarnation of Fontaine-Laffaille theory. Second of all, since is by definition the kernel of the map
we get a complex
Taking Galois invariants gives a connecting homomorphism
in Galois cohomology, a primordial Bloch-Kato exponential. On the other hand, we have the sequence
of -modules. Passing to -invariants and taking the inverse limit over induces the more familiar Kummer connecting homomorphism
in Galois cohomology, and it turns out that . In fact this is pretty immediate from the diagram
where the middle downwards arrow sends to and the rightmost downwards arrow is . (Sorry for the terrible formatting!) This comparison vividly illustrates why lands in .
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 converges to a well-defined element of . Furthermore, .
Step Two. For any , .
Step Three. The image of in is independent of the choice of , and depends only on the equivalence class of .
The result is immediate from here: the first two steps show is a Cauchy sequence in the -adic topology on , and the third step shows that its limit has the claimed properties.
Proof of Step One. In fact, something more general is true: if converges on the open unit disk, and is any element with , then converges in . To see this, note that the topology on is defined by a natural p-adic valuation , with equal to the greatest integer such that . Under our hypotheses, we may choose some large with , which is to say with . The latter ideal certainly has divided powers in , so , which implies in particular that for some constant . Since by hypothesis for any , the claimed convergence follows.
For the more delicate second statement, note that by our assumptions we have and so has divided powers in . Writing
the claim then follows from the fact that .
Proof of Step Two. Since is a differential of the second kind, we have
so an easy induction gives
for some . In a similar vein, we have
for some . Using these identities, we rewrite the difference in question as follows:
It’s easy to see the – and -terms here converge in , so all that remains is to show . Now something great happens. Writing , we have
so ! Therefore has divided powers in and defines an element of exactly as step one.
Proof of Step Three.If and are two different lifts, then
(with the notation as in step two). Since , the right-hand side of this expression gives a well-defined element of , so the independence-of-the-lift claim follows. Likewise, if , then
for some , and we’re done.