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Let $K$ be a finite extension of $\mathbb{Q}_p$ and $L/K$ a finite unramified extension. Let $M$ be a $(\phi, \Gamma_L)$-module over the Robba ring of $L$ (with coefficients in some other $p$-adic field $E$).

If $M$ is trianguline then is $\mathrm{Ind}^L_K(M)$ also trianguline?

Since $L/K$ is unramified, $\mathrm{Ind}^L_K(M)$ is simply $M$ with the same $\phi$ and $\Gamma$ actions but viewed as a module over the Robba ring of $K$. Also, this reduces trivially to the case that $M$ has rank $1$.

I am mostly interested in the etale case, but the above question seems more natural. The motivation is that similar statements are true for crystalline or semi-stable (etale) $(\phi, \Gamma)$-modules, so it seems natural to expect similar behaviour in the trianguline case.

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The answer is NO in general. Laurent Berger studies in this paper:

http://perso.ens-lyon.fr/laurent.berger/articles/article18.pdf

inductions of 1-dimensional representations of the absolute Galois group of $\mathbb Q_{p^2}$ (these are always trianguline) to the Galois group of $\mathbb Q_p$. The resulting induction are not always trianguline.

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  • $\begingroup$ Thanks a lot for the reference! $\endgroup$
    – naf
    Commented Jan 9, 2022 at 13:42

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