Iwasawa theory of de Rham $(\varphi , \Gamma )$-modules over the Robba ring

• Published in: Journal of the Institute of Mathematics of Jussieu Vol. 13; no. 1; pp. 65 - 118
• Main Author: Nakamura, Kentaro
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.01.2014
• Subjects: Algebra; Applied mathematics; Construction; Geometry; Secondary 11F85; 11S25; B-pair; Primary 11F80; p-adic Hodge theory; (φ,Γ)-module
• ISSN: 1474-7480; 1475-3030
• DOI: 10.1017/S1474748013000078

The aim of this article is to study the Bloch–Kato exponential map and the Perrin-Riou big exponential map purely in terms of $(\varphi , \Gamma )$-modules over the Robba ring. We first generalize the definition of the Bloch–Kato exponential map for all the $(\varphi , \Gamma )$-modules without using Fontaine’s rings ${\mathbf{B} }_{\mathrm{crys} } $, ${\mathbf{B} }_{\mathrm{dR} } $ of $p$-adic periods, and then generalize the construction of the Perrin-Riou big exponential map for all the de Rham $(\varphi , \Gamma )$-modules and prove that this map interpolates our Bloch–Kato exponential map and the dual exponential map. Finally, we prove a theorem concerning the determinant of our big exponential map, which is a generalization of theorem $\delta (V)$ of Perrin-Riou. The key ingredients for our study are Pottharst’s theory of the analytic Iwasawa cohomology and Berger’s construction of $p$-adic differential equations associated to de Rham $(\varphi , \Gamma )$-modules.