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I have a, hopefully not so stupid, feeling that the notion of $(\phi,\Gamma)$-modules is an abstract model for $p$-adic Galois representations.

I am wondering what is good for us when we work with such notion. For example, is there something being easy when one works with $(\phi,\Gamma)$-modules instead of $p$-adic Galois representations?

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    $\begingroup$ It’s much more linear algebraic in nature than the notion of a p-adic Galois rep. $\endgroup$ Commented Sep 16 at 12:44
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    $\begingroup$ Also, just like with any other non-trivial classification, you can find structures and properties on one side that are much more visible than on the other. For instance, here you have the notion of a trianguline Galois representation, which is most easily defined using the theory of $(\varphi,\Gamma)$-modules. $\endgroup$ Commented Sep 16 at 15:19
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    $\begingroup$ You can also write down explicitly the data determining an extension class of (phi, Gamma) modules much more easily than you can for Galois representations, so you get a much simpler description of $H^1(K, V)$. $\endgroup$ Commented Sep 17 at 8:56

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