Coates–wiles homomorphisms and iwasawa cohomology for lubin–tate extensions

Abstract

For the p-cyclotomic tower of Qp Fontaine established a description of local Iwasawa cohomology with coefficients in a local Galois representation V in terms of the ψ-operator acting on the attached etale (φ, Γ)-module D(V). In this chapter we generalize Fontaine's result to the case of arbitrary Lubin–Tate towers L∞ over finite extensions L of Qp by using the Kisin–Ren/Fontaine equivalence of categories between Galois representations and (φL, ΓL)-modules and extending parts of [20, 33]. Moreover, we prove a kind of explicit reciprocity law which calculates the Kummer map over L∞ for the multiplicative group twisted with the dual of the Tate module T of the Lubin–Tate formal group in terms of Coleman power series and the attached (φL, ΓL)-module. The proof is based on a generalized Schmid–Witt residue formula. Finally, we extend the explicit reciprocity law of Bloch and Kato [3] Theorem 2.1 to our situation expressing the Bloch–Kato exponential map for L(Math Presented) in terms of generalized Coates–Wiles homomorphisms, where the Lubin–Tate character χLT describes the Galois action on T.