The weight-monodromy conjecture for motives

Let $M$ be a motive of weight $w=i-2j$, let $v\nmid p$ be a finite place of $F$, and let $\sigma(v)$ be a lift of $\operatorname{Fr}(v)$ to $G_{F_v}$. Consider the semisimple Weil–Deligne representation attached to the $G_{F_v}$-action on $M_{\operatorname{et},\mathfrak p}$ for $\mathfrak p\mid p$.

Weight-monodromy conjecture. This representation is pure of weight $w$; namely, the eigenvalues of $\sigma(v)$ on the $r$-th graded piece of the monodromy filtration are Weil numbers of weight $w+r$.

The conjecture is a central expected property of the realizations of motives and is used in the paper to define motivic Galois representations.