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For the p-cyclotomic tower of \(\mathbb {Q}_p\) Fontaine established a description of local Iwasawa cohomology with coefficients in a local Galois representation V in terms of the \(\psi \)-operator acting on the attached etale \((\varphi ,\Gamma )\)-module D(V). In this chapter we generalize Fontaine’s result to the case of arbitrary Lubin–Tate towers \(L_\infty \) over finite extensions L of \(\mathbb {Q}_p\) by using the Kisin–Ren/Fontaine equivalence of categories between Galois representations and \((\varphi _L,\Gamma _L)\)-modules and extending parts of [20, 33]. Moreover, we prove a kind of explicit reciprocity law which calculates the Kummer map over \(L_\infty \) 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 \((\varphi _L,\Gamma _L)\)-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 \(L(\chi _{LT}^r)\) in terms of generalized Coates–Wiles homomorphisms, where the Lubin–Tate character \(\chi _{LT}\) describes the Galois action on T.

Coates–Wiles Homomorphisms and Iwasawa Cohomology for Lubin–Tate Extensions