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 $G\_K$?



If the tensor product of two representations are crystalline, are the original representations crystalline?
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.


