On Perrin-Riou's exponential map for $(\varphi, \Gamma)$-modules

TL;DR

This paper generalizes Perrin-Riou's exponential map for $(, )$-modules over the Robba ring, connecting Galois cohomology with $B$-pairs and interpolating exponential maps across cyclotomic extensions.

Contribution

It introduces a new big exponential map $o_{D,h}$ for $(, )$-modules, extending the Bloch-Kato exponential to a broader setting using Berger's $B$-pairs.

Findings

Constructed a generalized exponential map for $(, )$-modules.

Interpolates exponential maps across cyclotomic extensions.

Links Galois cohomology with $B$-pair theory.

Abstract

Let $K / \mathbb{Q}_p$ be a finite Galois extension and $D$ a $(\varphi, \Gamma)$-module over the Robba-ring $B^{\dagger}_{\textrm{rig}, K}$. We give a generalization of the Bloch-Kato exponential map for $D$ using continuous Galois-cohomology groups $H^i(G_K, W(D))$ for the $B$-pair $W(D)$ associated to $D$. We construct a big exponential map $\Omega_{D,h}$ ($h \in \mathbb{N}$) for cyclotomic extensions of $K$ for $D$ in the style of Perrin-Riou using the theory of Berger's $B$-pairs, which interpolates the generalized Bloch-Kato exponential maps on the finite levels.