## A simple construction of Sen’s functor

Let ${p}$ be a prime, and let ${G_{\mathbf{Q}_{p}}=\mathrm{Gal}(\overline{\mathbf{Q}_{p}}/\mathbf{Q}_{p})}$. A ${\mathbf{C}_{p}}$semilinear representation of ${G_{\mathbf{Q}_{p}}}$ is a finite-dimensional ${\mathbf{C}_{p}}$-vector space ${W}$ with a continuous action of ${G_{\mathbf{Q}_{p}}}$ such that ${g(aw)=g(a)g(w)}$ for all ${a\in\mathbf{C}_{p}}$, ${w\in W}$ and ${g\in G_{\mathbf{Q}_{p}}}$. One of the early successes of p-adic Hodge theory was Sen’s classification of ${\mathbf{C}_{p}}$-semilinear representations: given such a representation ${W}$, say with ${\mathrm{dim}_{\mathbf{C}_{p}}W=n}$, Sen defined an ${n}$-dimensional ${\mathbf{Q}_{p}(\zeta_{p^{\infty}})}$-vector space ${\mathbf{D}_{\mathrm{Sen}}(W)}$ together with a ${\mathbf{Q}_{p}(\zeta_{p^{\infty}})}$linear endomorphism ${\Theta}$ such that ${W}$ is determined up to isomorphism by the pair ${(\mathbf{D}_{\mathrm{Sen}}(W),\Theta)}$. If ${V}$ is a representation of ${G_{\mathbf{Q}_{p}}}$ on a finite-dimensional ${\mathbf{Q}_{p}}$-vector space (“a p-adic representation”), then extending scalars up to ${\mathbf{C}_{p}}$ and setting ${\mathbf{D}_{\mathrm{Sen}}(V)=\mathbf{D}_{\mathrm{Sen}}(V\otimes_{\mathbf{Q}_{p}}\mathbf{C}_{p})}$, we see that ${V\otimes_{\mathbf{Q}_{p}}\mathbf{C}_{p}}$ 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 ${\mathbf{Q}_{p}}$-algebras ${\mathbf{B}_{\bullet}}$, ${\bullet\in\left\{ \mathrm{HT},\mathrm{dR},\mathrm{crys},\mathrm{st},...\right\} }$, each equipped with with a linear action of ${G_{\mathbf{Q}_{p}}}$ and some ${G_{\mathbf{Q}_{p}}}$-equivariant auxiliary structure (e.g. a filtration, Frobenius, monodromy operator…): the assignment

$\displaystyle V\rightsquigarrow\mathbf{D}_{\bullet}(V)=(V\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\bullet})^{G_{\mathbf{Q}_{p}}}$

is then some nice functor from Galois representations to modules over the ring ${\mathbf{B}_{\bullet}^{G_{\mathbf{Q}_{p}}}}$, and ${\mathbf{D}_{\bullet}(V)}$ inherits whatever extra structures ${\mathbf{B}_{\bullet}}$ had.

My goal in this post is to give a simple Fontaine-style description of ${\mathbf{D}_{\mathrm{Sen}}(V)}$. 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 ${H=\mathrm{Gal}(\overline{\mathbf{Q}_{p}}/\mathbf{Q}_{p}(\zeta_{p^{\infty}}))}$ and ${\Gamma=\mathrm{Gal}(\mathbf{Q}_{p}(\zeta_{p^{\infty}})/\mathbf{Q}_{p})\simeq G_{\mathbf{Q}_{p}}/H}$, so the cyclotomic character ${\chi}$ identifies ${\Gamma}$ with ${\mathbf{Z}_{p}^{\times}}$. Let ${\mathbf{B}_{\mathrm{an}}=\mathscr{C}_{\mathrm{an}}(\mathbf{Z}_{p}^{\times},\mathbf{C}_{p})}$ be the ring of locally analytic ${\mathbf{C}_{p}}$-valued functions on ${\mathbf{Z}_{p}^{\times}}$, with the continuous Galois action defined by

$\displaystyle (g\star f)(x)=g\cdot(f(\chi(g^{-1})x))$

for ${x\in\mathbf{Z}_{p}^{\times}}$ and ${g\in G_{\mathbf{Q}_{p}}}$. This ring is an inductive limit of Banach spaces. Note that ${\mathbf{B}_{\mathrm{an}}}$ also has a residual action of ${\Gamma}$ given by ${(\gamma f)(x)=f(\chi(\gamma)x)}$, and this action commutes with the Galois action.

Theorem. We have ${\mathbf{B}_{\mathrm{an}}^{G_{\mathbf{Q}_{p}}}=\mathbf{Q}_{p}(\zeta_{p^{\infty}})}$, and there is a canonical isomorphism

$\displaystyle \mathbf{D}_{\mathrm{Sen}}(V)\cong(V\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{an}})^{G_{\mathbf{Q}_{p}}}$

of ${\mathbf{Q}_{p}(\zeta_{p^{\infty}})}$-vector spaces equivariant for the residual actions of ${\Gamma}$ and identifying ${\Theta}$ with the operator ${1\otimes x\frac{d}{dx}}$.

Let’s warm up by calculating ${\mathbf{B}_{\mathrm{an}}^{G_{\mathbf{Q}_{p}}}}$.

Fact: ${\mathbf{B}_{\mathrm{an}}^{H}=\mathscr{C}_{\mathrm{an}}(\mathbf{Z}_{p}^{\times},\widehat{\mathbf{Q}_{p}(\zeta_{p^{\infty}})})}$.

Proof. The subgroup ${H}$ acts trivially on the coordinate variable ${x}$, so ${f}$ defines an element of ${\mathbf{B}_{\mathrm{an}}^{H}}$ if and only if ${f(x)\in\mathbf{C}_{p}^{H}}$ for all ${x\in\mathbf{Z}_{p}^{\times}}$. Now use Ax-Sen-Tate.

Fact: ${(\mathbf{B}_{\mathrm{an}})^{G_{\mathbf{Q}_{p}}}=\mathbf{Q}_{p}(\zeta_{p^{\infty}})}$.

Proof. If ${f\in\mathbf{B}_{\mathrm{an}}^{G_{\mathbf{Q}_{p}}}}$ then in particular ${f\in\mathbf{B}_{\mathrm{an}}^{H}}$, so ${f(1)\in\widehat{\mathbf{Q}_{p}(\zeta_{p^{\infty}})}}$. Furthermore for any ${x\in\mathbf{Z}_{p}^{\times}}$ we have

$\displaystyle \begin{array}{rcl} f(1) & = & (\chi^{-1}(x^{-1})\star f)(1)\\ & = & \chi^{-1}(x^{-1})\cdot(f(x)),\end{array}$

so ${f(x)=\chi^{-1}(x)\cdot f(1)}$. Thus we may identify ${\mathbf{B}_{\mathrm{an}}^{G_{\mathbf{Q}_{p}}}}$ with the space of elements ${a\in\widehat{\mathbf{Q}_{p}(\zeta_{p^{\infty}})}}$ whose orbit function ${\gamma\cdot a}$ is locally analytic. The result now follows from Théorème 1.2.2 of [BC13].

Proof of the theorem. For brevity we write ${V_{\infty}=(V\otimes_{\mathbf{Q}_{p}}\mathbf{C}_{p})^{H}}$; this is an n-dimensional ${\widehat{\mathbf{Q}_{p}(\zeta_{p^{\infty}})}}$-vector space with a semilinear action of ${\Gamma}$. We have a natural identification

$\displaystyle \begin{array}{rcl} V\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{an}} & = & V\otimes_{\mathbf{Q}_{p}}\mathbf{C}_{p}\otimes_{\mathbf{C}_{p}}\mathscr{C}_{\mathrm{an}}(\mathbf{Z}_{p}^{\times},\mathbf{C}_{p})\\ & = & \mathscr{C}_{\mathrm{an}}(\mathbf{Z}_{p}^{\times},V\otimes_{\mathbf{Q}_{p}}\mathbf{C}_{p}),\end{array}$

so passing to ${H}$-invariants gives

$\displaystyle \begin{array}{rcl} \left(V\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{an}}\right)^{H} & = & \mathscr{C}_{\mathrm{an}}\left(\mathbf{Z}_{p}^{\times},V_{\infty}\right)\end{array}$

as ${H}$ acts trivially on the ${x}$-variable. Passing then to ${\Gamma}$-invariants, we identify ${(V\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{an}})^{G_{\mathbf{Q}_{p}}}}$ with the space of locally analytic functions ${f:\mathbf{Z}_{p}^{\times}\rightarrow V_{\infty}}$ such that ${f(\chi(\gamma))=\gamma\cdot f(1)}$. In particular, the evaluation map

$\displaystyle \begin{array}{rcl} (V\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{an}})^{G_{\mathbf{Q}_{p}}} & \rightarrow & V_{\infty}\\ f & \mapsto & f(1)\end{array}$

is injective and ${\Gamma}$-equivariant.

Now, recall that a vector ${v\in V_{\infty}}$ is ${\mathbf{Q}_{p}}$finite if ${v}$ lies in a sub-vector space of ${V_{\infty}}$ which is finite-dimensional over ${\mathbf{Q}_{p}}$ and stable under ${\Gamma}$; ${\mathbf{D}_{\mathrm{Sen}}(V)}$ is defined as the set of all ${\mathbf{Q}_{p}}$-finite vectors in ${V_{\infty}}$ with its natural ${\mathbf{Q}_{p}(\zeta_{p^{\infty}})}$-vector space structure. The key result we need is again Théorème 1.2.2 of [BC13]: a vector ${v\in V_{\infty}}$ is ${\mathbf{Q}_{p}}$-finite if and only if the orbit function ${f_{v}(g)=g\cdot v}$ is locally analytic. With this in hand, we see that the evaluation map defined above has image contained in ${\mathbf{D}_{\mathrm{Sen}}(V)}$, and the map

$\displaystyle \begin{array}{rcl} \mathbf{D}_{\mathrm{Sen}}(V) & \rightarrow & (V\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{an}})^{G_{\mathbf{Q}_{p}}}\\ v & \mapsto & f_{v}(\chi(\gamma))=\gamma\cdot v\end{array}$

gives a well-defined injective inverse.

It remains to calculate the action of ${\Theta}$, which is defined as ${\frac{\log\gamma}{\log\chi(\gamma)}}$ for any nontrivial element ${\gamma\in\Gamma}$ with ${\chi(\gamma)\equiv1\,\mathrm{mod}\, p}$. Using the formula ${\log x=\lim_{a\rightarrow0}\frac{x^{a}-1}{a}}$, it’s easy to see that ${\Theta}$ is given equally well by ${\lim_{n\rightarrow\infty}\frac{\gamma_{n}-1}{p^{n}}}$, where ${\gamma_{n}\in\Gamma}$ is the element with ${\chi(\gamma_{n})=1+p^{n}}$. Since the map ${v\mapsto f_{v}}$ is ${\mathbf{Q}_{p}}$-linear, an easy unwinding gives

$\displaystyle f_{\frac{\gamma_{n}-1}{p^{n}}v}(x)=\frac{f_{v}(x(1+p^{n}))-f_{v}(x)}{p^{n}},$

so inserting the Taylor expansion ${f_{v}(x+y)=f_{v}(x)+f_{v}'(x)y+O(y^{2})}$ and taking the limit as ${n\rightarrow\infty}$ gives

$\displaystyle \begin{array}{rcl} f_{\Theta v}(x) & = & \lim_{n\rightarrow\infty}\frac{f_{v}(x)+f_{v}'(x)p^{n}x+O(p^{2n})-f_{v}(x)}{p^{n}}\\ & = & \lim_{n\rightarrow\infty}xf_{v}'(x)+O(p^{n})\\ & = & xf_{v}'(x)\end{array}$

as desired.

$\displaystyle$

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.