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Iwasawa theory of de Rham
$(\varphi , \Gamma )$-modules over the Robba ring
- Part of
-
- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 13 / Issue 1 / January 2014
- Published online by Cambridge University Press:
- 27 February 2013, pp. 65-118
- Print publication:
- January 2014
-
- Article
- Export citation
-
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.
