#### Vol. 11, No. 2, 2017

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A generalization of Kato's local $\varepsilon$-conjecture for $(\varphi,\Gamma)$-modules over the Robba ring

### Kentaro Nakamura

Vol. 11 (2017), No. 2, 319–404
##### Abstract

We generalize Kato’s (commutative) $p$-adic local $\epsilon$-conjecture for families of $\left(\phi ,\Gamma \right)$-modules over the Robba rings. In particular, we prove the essential parts of the generalized local $\epsilon$-conjecture for families of trianguline $\left(\phi ,\Gamma \right)$-modules. The key ingredients are the author’s previous work on the Bloch–Kato exponential map for $\left(\phi ,\Gamma \right)$-modules and the recent results of Kedlaya, Pottharst and Xiao on the finiteness of cohomology of $\left(\phi ,\Gamma \right)$-modules.

##### Keywords
$p$-adic Hodge theory, $(\varphi,\Gamma)$-module, $B$-pair
##### Mathematical Subject Classification 2010
Primary: 11F80
Secondary: 11F85, 11S25