$ \epsilon $-ISOMORPHISMS FOR RANK ONE $( \varphi , \Gamma )$-MODULES OVER LUBIN-TATE ROBBA RINGS

Abstract Inspired by Nakamura’s work [36] on $\epsilon $ -isomorphisms for $(\varphi ,\Gamma )$ -modules over (relative) Robba rings with respect to the cyclotomic theory, we formulate an analogous conjecture for L -analytic Lubin-Tate $(\varphi _L,\Gamma _L)$ -modules over (relative) Robba rings for any finite extension L of $\mathbb {Q}_p.$ In contrast to Kato’s and Nakamura’s setting, our conjecture involves L -analytic cohomology instead of continuous cohomology within the generalized Herr complex. Similarly, we restrict to the identity components of $D_{cris}$ and $D_{dR},$ respectively. For rank one modules of the above type or slightly more generally for trianguline ones, we construct $\epsilon $ -isomorphisms for their Lubin-Tate deformations satisfying the desired interpolation property.