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Let $p$ be a prime number. There are properties called ``overconvergence''and ``$F$-analyticity'' for $p$-adic Galois representations of a $p$-adic field$F$. By Berger's work, it is known that $F$-analyticity is stricter thanoverconvergence. In this article, we show that, in many cases, anoverconvergent Galois representation is $F$-analytic up to a twist by acharacter. This result emphasizes the necessity of the theory of$(\varphi,\Gamma)$-modules over the multivariable Robba ring, by which weexpect to study all $p$-adic Galois representations.

On a relation of overconvergence and $F$-analyticity on $p$-adic Galois representations of a $p$-adic field $F$