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 .
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.
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