Fontaine–Mazur–Langlands conjecture for geometric Galois representations

Let $F$ be a number field and let

$$\rho\colon\Gamma_F\to\mathrm{GL}_n(\overline{\mathbb{Q}}_\ell)$$

be an irreducible geometric Galois representation, meaning that it is almost everywhere unramified and de Rham at every place above $\ell$. Fontaine–Mazur–Langlands conjecture. The representation $\rho$ is motivic: it occurs as a subquotient of

$$H^j(X_{\overline{F}},\overline{\mathbb{Q}}_\ell)(r)$$

for some smooth projective variety $X/F$ and integer $r$, cut out by $\overline{\mathbb{Q}}_\ell$-linear combinations of homological algebraic cycles. Moreover, there is a cuspidal automorphic representation $\Pi$ of $\mathrm{GL}_n(\mathbf{A}_F)$ with the stated Frobenius–Satake and archimedean/Hodge–Tate compatibilities, and conversely every cuspidal $\Pi$ with integral archimedean Langlands parameters has a corresponding irreducible geometric representation $\rho_\Pi$. This combines the Fontaine–Mazur and Langlands expectations; the source states it as a conjectural framework and does not resolve it in general.