2015/11/16 by Peter Schneider, Schneider, Peter, Otmar Venjakob +1
Mathematics · #11Sxx #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #math.NT #msc:11Sxx
paper · pdf · doi:10.48550/arxiv.1511.04922
54 pages
arxiv created 2015/11/16 · openalex publication_date 2015/11/16 · arxiv updated 2015/11/17 · openalex created_date 2022/10/07 · openalex updated_date 2026/07/28
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 article we generalize Fontaine's result to the case of arbitratry 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)-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 (ϕ,Γ)-modules. Documenta Math. 2006, Extra Vol., 685-709]. 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 [Bloch S., Kato K.: L-functions and Tamagawa numbers of motives. The Grothendieck Festschrift, Vol. I, 333-400, Progress Math., 86, Birkhäuser Boston 1990] Thm. 2.1 to our situation expressing the Bloch-Kato exponential map for L(χLTr) in terms of generalized Coates-Wiles homomorphisms, where the Lubin-Tate characater χLT describes the Galois action on T.