Fontaine–Mazur conjecture

Fix a prime $p$ and an algebraic closure $\overline{\mathbb{Q}}$ of $\mathbb{Q}$, and let $G_{\mathbb{Q}} = \mathrm{Gal}(\overline{\mathbb{Q}}\,|\,\mathbb{Q})$. Let $\chi_p \colon G_{\mathbb{Q}} \to \mathbb{Z}_p^{\times}$ be the $p$-adic cyclotomic character, and let $c \in G_{\mathbb{Q}}$ denote a complex conjugation (the image of the nontrivial element of $\mathrm{Gal}(\mathbb{C}|\mathbb{R})$ under an embedding $\overline{\mathbb{Q}} \hookrightarrow \mathbb{C}$).

Let

$$\rho \colon G_{\mathbb{Q}} \longrightarrow \mathrm{GL}_2(\overline{\mathbb{Q}}_p)$$

be a continuous irreducible representation satisfying:

1. $\rho$ is unramified at all but finitely many primes $\ell$; 2. there is no integer $n$ and no representation $\rho_0 \colon G_{\mathbb{Q}} \to \mathrm{GL}_2(\overline{\mathbb{Q}}_p)$ factoring through a finite quotient of $G_{\mathbb{Q}}$ with $\det \rho_0(c) = 1$ (an even representation of finite image) such that $\rho \cong \rho_0 \otimes \chi_p^{\,n}$.

Say that $\rho$ is potentially semi-stable at $p$ if the restriction of $\rho$ to a decomposition group $G_{\mathbb{Q}_p} = \mathrm{Gal}(\overline{\mathbb{Q}}_p\,|\,\mathbb{Q}_p) \subseteq G_{\mathbb{Q}}$ at $p$ becomes semi-stable, in the sense of Fontaine's theory of $p$-adic Hodge structures, after restriction to $G_K$ for some finite extension $K/\mathbb{Q}_p$ inside $\overline{\mathbb{Q}}_p$; equivalently, $\rho|_{G_{\mathbb{Q}_p}}$ is de Rham.

Say that $\rho$ is associated to a cuspidal newform if there exist integers $k \ge 2$, $N \ge 1$, a normalized cuspidal newform $f = \sum_{n \ge 1} a_n(f) q^n$ of weight $k$ on $\Gamma_1(N)$, and an embedding of the field generated by the $a_n(f)$ into $\overline{\mathbb{Q}}_p$, such that for every prime $\ell \nmid Np$ at which $\rho$ is unramified,

$$\mathrm{tr}\,\rho(\mathrm{Frob}_{\ell}) = a_{\ell}(f),$$

i.e. $\rho$ is isomorphic to the $p$-adic Galois representation attached to $f$ up to twist by an integral power of $\chi_p$.

Then $\rho$ is associated to a cuspidal newform if and only if $\rho$ is potentially semi-stable at $p$.