On Perrin-Riou's exponential map and reciprocity laws for (phi,Gamma)-modules

Abstract

Let K / Q_p be a finite Galois extension and D a (phi,Gamma)-module over the Robba ring B and N_dR(D) its associated p-adic differential equation. In the first part 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 for cyclotomic extensions of K for D in the style of Perrin-Riou extending techniques of Berger, which interpolates the generalized Bloch-Kato exponential maps on the finite levels. In the second part we extend two definitions for pairings on D and its dual D^*(1) (resp. on N_dR(D) and its dual N_dR(D^*(1))) and prove a generalization of the reciprocity law, which relates these pairings under the big exponential map. Finally, we give some results on the determinant associated to Omega_D,h, and formulate an integral version of a determinant conjecture in the semi-stable case. Further, we define i-Selmer groups and show under certain hypothesis a torsion property.