Let be a prime, and let . A –*semilinear representation of * is a finite-dimensional -vector space with a continuous action of such that for all , and . One of the early successes of *p-*adic Hodge theory was Sen’s classification of -semilinear representations: given such a representation , say with , Sen defined an -dimensional -vector space together with a –*linear* endomorphism such that is determined up to isomorphism by the pair . If is a representation of on a finite-dimensional -vector space (“a *p-*adic representation”), then extending scalars up to and setting , we see that is determined by some incredibly simple data in which all the intricacies of the Galois action are gone.

This result exemplifies a basic precept of *p-*adic Hodge theory: *p-*adic representations can be meaningfully analyzed and classified by way of functors which map Galois representations to “modules with extra structure” over auxiliary rings. Fontaine codified this by defining a slew of topological -algebras , , each equipped with with a linear action of and some -equivariant auxiliary structure (e.g. a filtration, Frobenius, monodromy operator…): the assignment

is then some nice functor from Galois representations to modules over the ring , and inherits whatever extra structures had.

My goal in this post is to give a simple Fontaine-style description of . Maybe surprisingly, this is a new result; the main ingredient is an important recent theorem of Berger and Colmez (reference below). As usual, we write and , so the cyclotomic character identifies with . Let be the ring of locally analytic -valued functions on , with the continuous Galois action defined by

for and . This ring is an inductive limit of Banach spaces. Note that also has a residual action of given by , and this action commutes with the Galois action.

**Theorem. ***We have , and there is a canonical isomorphism*

*of -vector spaces equivariant for the residual actions of and identifying with the operator .*

Let’s warm up by calculating .

**Fact**: .

*Proof. *The subgroup acts trivially on the coordinate variable , so defines an element of if and only if for all . Now use Ax-Sen-Tate.

**Fact**: .

*Proof. *If then in particular , so . Furthermore for any we have

so . Thus we may identify with the space of elements whose orbit function is locally analytic. The result now follows from Théorème 1.2.2 of [BC13].

*Proof of the theorem. *For brevity we write ; this is an *n*-dimensional -vector space with a semilinear action of . We have a natural identification

so passing to -invariants gives

as acts trivially on the -variable. Passing then to -invariants, we identify with the space of locally analytic functions such that . In particular, the evaluation map

is injective and -equivariant.

Now, recall that a vector is –*finite* if lies in a sub-vector space of which is finite-dimensional over and stable under ; is defined as the set of all -finite vectors in with its natural -vector space structure. The key result we need is again Théorème 1.2.2 of [BC13]: a vector is -finite if and only if the orbit function is locally analytic. With this in hand, we see that the evaluation map defined above has image contained in , and the map

gives a well-defined injective inverse.

It remains to calculate the action of , which is defined as for any nontrivial element with . Using the formula , it’s easy to see that is given equally well by , where is the element with . Since the map is -linear, an easy unwinding gives

so inserting the Taylor expansion and taking the limit as gives

as desired.

This is a very nice interpretation. We can probably argue similarly with Fontaine’s construction of $D_{diff}(V)$ to show that $D_{diff}(V) = (V \otimes_{\mathbf{Q}_p} \mathbf{B}_{dR, diff})^{\mathcal{G}_{\mathcal{Q}_p}}$ where $\mathbf{B}_{dR, diff} = \mathcal{C}_{an}(\mathbf{Z}_p^\times, \mathbf{B}_{dR})$ and again explicitly calculate the connection.

Yes, I had wondered about this! There is a slight extra wrinkle in that B_dR^{+} is “merely” Frechet, as opposed to Banach…