Conductor of Galois representation attached to newform

Question

(Sorry for poor my english skill..)

Let $k$ and $N$ be positive integers and $\chi$ be a Dirichlet character modulo $N$. Let $F$ be a newform with number field $K_{F}$. (All coefficients of $F$ in $K_{F}$.) Let $\ell$ be a prime and let $\lambda$ of $\mathcal{O}_{K_{F}}$ be a maximal ideal lying over $\ell$. By theorem 9.6.5 in "A first course in Modular forms - Diamond and Shurman", there is an irreducible 2-dimensional Galois representation \begin{equation} \rho_{F,\lambda} : G_{\mathbb{Q}} \to GL_2(K_{F,\lambda}). \end{equation} Someone told me that the conductor of $\rho_{F,\lambda}$ is same as the level of $F$ i.e. the conductor of $\rho_{F,\lambda}$ is $N$. Also he said that this fact is in the Carayol's paper, however I couldn't find it.

Is the fact true? If it is true, I would appreciate your reference.

Thanks for reading.

Answer 1

In fact much more than the equality of conductor is true: the local Galois representation $\rho_{F,\lambda}|G_{\mathbb Q_{p}}$ obtained by restricting $\rho_{F,\lambda}$ to the decomposition group at $p$ corresponds in a precise way to the local automorphic representation $\pi(F)_{p}$. This is the so-called local-global compatibility property of the Langlands reciprocity conjectures.

This is indeed a theorem of Henri Carayol* and here follows a precise reference.

Sur les représentations $l$-adiques associées aux formes modulaires de Hilbert Carayol, Henri. Annales scientifiques de l'É.N.S tome 19 n°3 (1986) page 409-468.

The relevant theorem is Théorème (A) page 410.

*To be precise, Carayol's work goes through a compatible system of Galois representation, if you want to relate the single Galois representation $\rho_{F,\lambda}$ directly to $\pi(F)_\ell$ (so if you want to the power of $\ell$ appearing in the conductor and level without appealing to the compatible system attached to $F$), then you need a result of Takeshi Saito (Inventiones mathematicae,1997).