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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2$\begingroup$ Is the zerodimensional representation crystalline? If so....(Sorry for this.) $\endgroup$ – Pete L. Clark Apr 29 '10 at 18:37

$\begingroup$ 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. $\endgroup$ – Hunter Brooks Apr 29 '10 at 18:39

3$\begingroup$ @Hunter: the problem is that V and W might not even be HodgeTate! Consider for example a random 1dimensional nonHodgeTate V and let W be its dual! $\endgroup$ – Kevin Buzzard Apr 29 '10 at 19:40

1$\begingroup$ @FC: presumably you can do the elladic 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 HodgeTate weights at least. $\endgroup$ – Kevin Buzzard Apr 29 '10 at 19:44

1$\begingroup$ @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. $\endgroup$ – Kevin Buzzard Apr 30 '10 at 0:05
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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.

2$\begingroup$ Just out of curiosity, had you already asked your student to do this before the question came up on MO? $\endgroup$ – Ben Webster♦ May 1 '10 at 19:55

5$\begingroup$ I asked my student to do this in December of 2009. As far as I can remember, it was Barry Mazur who asked me this question when I was at Harvard (so that was at least 5 or 6 years ago). If I remember correctly, he wanted to know what "Sym^2 V crystalline" implied about V. Between then and now, a couple more people asked me the more general question about V \otimes W (I unfortunately don't remember their names). In both cases, I told them the method which I thought would solve the problem, and didn't hear back from them. $\endgroup$ – Laurent Berger May 2 '10 at 8:40