# About the restriction of a modular representation to a decomposition subgroup

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 $p | N$.

• You have to specify the residue characteristic of the linear coefficients of $\rho_f$ to get a sensible answer. – jfb Jul 7 '14 at 12:25
• Isn't it the representation associated by the Local Langlands correspondence to an unramified principal series? – Ricky Jul 7 '14 at 15:21
• If $a_p(f)$ is a unit, then the representation is reducible, and the two Jordan--Holder factors are explicit (one is unramified, the other is unramified times a power of the cyclotomic). In the non-ordinary case there's not a lot you can say without invoking Fontaine's classfication of p-adic representations of $G_{\mathbf{Q}_p}$. – David Loeffler Jul 7 '14 at 16:24

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.

• TL;DR David Loeffler's and Ricky's comments are spot on. – Olivier Jul 8 '14 at 6:02
• "In principle again, this describes completely $V(\rho_{f,p})$" -- not quite! You have to specify the filtration too. If $a_p$ is a non-unit then there is a unique possibility up to isomorphism. If $a_p$ is a unit, then there are two possibilities: the nontrivial filtration subspace can coincide with the non-unit Frobenius eigenspace, or it can be in general position, corresponding to the case where $\rho_{f, p}$ is a direct sum, or a non-split extension, of two characters. – David Loeffler Jul 8 '14 at 6:44
• Ah! So the Weil-Deligne representation is not enough. Interesting. – Olivier Jul 8 '14 at 18:22