TITLE: If the tensor product of two representations are crystalline, are the original representations crystalline?
QUESTION [15 upvotes]: Let $K$ be a finite extension of the $p$-adic numbers. Suppose that $V$ and $W$ are two (finite dimensional, $p$-adic) continuous representations of $G_K$. Suppose that $V \otimes W$ is crystalline. Is $V$ crystalline up to twist by a character of $G_K$?

REPLY [17 votes]: I'm indeed pretty sure that the answer is "yes". I'd prefer not to post the idea of the proof here because I asked one of my PhD students to write it down with all the details.