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

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2 Responses to A simple construction of Sen’s functor

  1. user says:

    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.

    • arithmetica says:

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

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