Coates-Wiles homomorphisms and Iwasawa cohomology for Lubin-Tate extensions

Abstract

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 article we generalize Fontaine's result to the case of arbitratry 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 $(\varphi_L,\Gamma_L)$-module and extending parts of [Herr L.: Sur la cohomologie galoisienne des corps $p$-adiques. Bull. Soc. Math. France 126, 563-600 (1998)], [Scholl A. J.: Higher fields of norms and $(\phi,\Gamma)$-modules. Documenta Math.\ 2006, Extra Vol., 685-709]. 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 [Bloch S., Kato K.: $L$-functions and Tamagawa numbers of motives. The Grothendieck Festschrift, Vol. I, 333-400, Progress Math., 86, Birkh\"auser Boston 1990] Thm. 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 characater $\chi_{LT}$ describes the Galois action on $T.$