This is the converse question to an earlier question. More precisely,

Let $X/K$ be a curve(or variety) over a global field $K$. We consider the Galois representation obtained by the absolute Galois group $G_K$ acting on $H_{et}^i(X_{/\bar K}, \mathbb{Q}_\ell)$.

Do properties of this representation, such as "unramified at a place $v$", semistable, de Rham, crystalline, Hodge-Tate, and so on and so forth, imply some geometric properties about $X_{/K}$? (I must confess that I know the proper definition of only the first property in this list, but I nevertheless put in the whole list from the original question for good measure.)

If so, please give examples.