Iwasawa theory and $F$-analytic Lubin-Tate $(\varphi,\Gamma)$-modules

TL;DR

This paper extends Iwasawa theory and the construction of cohomology classes using $(, )$-modules in the Lubin-Tate setting, generalizing Perrin-Riou's period map for crystalline representations.

Contribution

It introduces corestriction-compatible classes in Galois cohomology using Lubin-Tate $(, )$-modules and generalizes Perrin-Riou's period map to this setting.

Findings

Constructed corestriction-compatible cohomology classes.

Explicit description of classes for crystalline representations.

Generalization of Perrin-Riou's period map to Lubin-Tate context.

Abstract

Let $K$ be a finite extension of $\mathbf{Q}_p$. We use the theory of $(\varphi,\Gamma)$-modules in the Lubin-Tate setting to construct some corestriction-compatible families of classes in the cohomology of $V$, for certain representations $V$ of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/K)$. If in addition $V$ is crystalline, we describe these classes explicitly using Bloch-Kato's exponential maps. This allows us to generalize Perrin-Riou's period map to the Lubin-Tate setting.