Abstract
AbstractThe 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.
Publisher
Cambridge University Press (CUP)
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