About the restriction of a modular representation to a decomposition subgroup

Question

Let $f$ be an eigenform of level $\Gamma_1(N)$ and let $p$ be a prime that does not divide $N$. It is well know that there is a $2$-dimensional representation $$ \rho_f \colon G_{\mathbb Q} \to \operatorname{GL}_2(E), $$ where $E$ is a finite extension of $\mathbb Q_p$, of the absolute Galois group of $\mathbb Q$ attached to $f$. We can restrict $\rho$ to a decomposition subgroup at $p$ (after having fixed the usual isomorphisms), obtaining a representation $\rho_{f,p}$ of the absolute Galois group of $\mathbb Q_p$.

Is it possible to give an explicit description of this representation? Note that I'm in the case $p$ does not divide $N$, that should be easier than the case $p | N$.

Answer 1

This hinges on what you mean exactly by explicit description. Here is what is happening. Let me write $N_f$ for the conductor of $f$.

Fontaine defined a number of so-called period rings to study $p$-adic representations of $G_{\mathbb Q_{p}}$ (from now on, $V$ is such a representation). One of these rings is called $B_{\operatorname{cris}}$. Say that $V$ is crystalline if the dimension of $D(V)=(B_{\operatorname{cris}}\otimes_{\mathbb Q_{p}}V)^{G_{\mathbb Q_{p}}}$ over $\mathbb Q_{p}$ is equal to the dimension of $V$. The first result towards an explicit description of $\rho_{f,p}$ is that $V(\rho_{f,p})$ is crystalline (this follows from you hypothesis that $p\nmid N_f$). This is a result of Scholl building on theorems of Fontaine-Messing and Faltings (among many others).

It turns out that when $V$ is crystalline, $D(V)$ is a so-called admissible filtered $\varphi$-module (the $\varphi$ indicates that $D(V)$ has an action of a Frobenius morphism) and in fact the category of crystalline representations and the category of admissible filtered $\varphi$-modules are equivalent (by the functor $D(\cdot)$) (this is a theorem of Colmez and Fontaine). So in principle, it is enough to describe $D(V(\rho_{f,p}))$ to know $V(\rho_{f,p})$ (in practice, whether this is satisfying to you depends on what you want to know exactly about $V(\rho_{f,p})$).

There are many things one can say about $D(V(\rho_{f,p}))$. The first is that the characteristic polynomial of $\varphi$ is equal to the Hecke polynomial $X^2-a_{p}X+\chi(p)p^{k-1}$ where as usual $a_p$ is the eigenvalue of $f$ under the Hecke operator $T(p)$, $\chi$ is the central character of $f$ and $k$ is its weight. More precisely, to $D(V)$ is attached a Weil-Deligne representation (a two-dimensional representation of the Weil group plus an action of a monodromy operator $N$) and this Weil-Deligne representation corresponds through the Local Langlands Correspondance to the unramified principal series representation $\pi(f)_{p}$ (again, that it is unramified principal series comes from the fact that $p\nmid N_f$) so in particular the monodromy $N$ is trivial (this is a result of Scholl using a theorem of Katz-Messing).

In principle again, this describes completely $V(\rho_{f,p})$ so the answer to your question is $V(\rho_{f,p})$ is the two-dimensional crystalline representation corresponding to the unramified principal series $\pi(f)_{p}$ through the LLC.