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

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 $G_K$?

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Is the zero-dimensional representation crystalline? If so....(Sorry for this.) – Pete L. Clark Apr 29 '10 at 18:37
Idea - we (i.e., not me) know the combinatorics of how tensoring two filtered phi modules affect the Hodge and Newton polygons, and we know the Hodge and Newton polygons of characters. So I think a proof or counterexample could be constructed by thinking about these pictures. – Hunter Brooks Apr 29 '10 at 18:39
@Hunter: the problem is that V and W might not even be Hodge-Tate! Consider for example a random 1-dimensional non-Hodge-Tate V and let W be its dual! – Kevin Buzzard Apr 29 '10 at 19:40
@FC: presumably you can do the ell-adic case? If V tensor W is unramified, is V a twist of an unramified rep? I am wondering whether you might want to start by looking at Sen operators and twisting so that V and W have integral Hodge-Tate weights at least. – Kevin Buzzard Apr 29 '10 at 19:44
@FC: if you just want to know the answer, it's "yes", and I know this because I asked Berger. He didn't tell me why though. – Kevin Buzzard Apr 30 '10 at 0:05