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$ \epsilon $-ISOMORPHISMS FOR RANK ONE 
$( \varphi , \Gamma )$-MODULES OVER LUBIN-TATE ROBBA RINGS
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- Journal:
- Journal of the Institute of Mathematics of Jussieu , First View
- Published online by Cambridge University Press:
- 11 April 2025, pp. 1-100
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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.
