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

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Elliptic Curves, Modular Forms and Iwasawa Theory (JHC70 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 188))

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 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.

To John Coates on the occasion of his 70th birthday.

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Notes

  1. 1.

    Note that, for any finite extension K / L contained in \(L_\infty \) the definition \(H^*_{\mathrm {Iw}}(L_\infty /K,V) := \varprojlim _{K \subseteq K' \subseteq L_\infty } H^*(K',V)\) produces the same \(o_L\)-modules. Our notation indicates that we always consider these groups as \(\Gamma _L\)-modules.

  2. 2.

    Another alternative formulation for the definition of \((\; ,\; )\) goes as follows: The residue pairing

    $$\begin{aligned} \mathrm {Res} : \mathscr {A}_L/\pi _L^n\mathscr {A}\times \Omega ^1_{\mathscr {A}_L/\pi _L^n\mathscr {A}}&\longrightarrow o_L/\pi _L^no_L \end{aligned}$$

    induces the pairing

    where the middle vertical map is induced by \(d\log \) and the inverse of the isomorphism \(o_L((Z))^\times /(1+\pi _Lo_L[[Z]])\cong K^\times \).

  3. 3.

    Setting \(L^r_{adm} := L \cap (\pi _L^{-r}\phi _q-1)(\,Fil^rB_{max,L}^+ )\) we still may define

    $$\begin{aligned} L^r_{adm} \xrightarrow {\partial ^r} H^1(L, L t_L^r) \end{aligned}$$

    without knowing the right hand surjectivity in Lemma 8.2.i and define \(\partial ^r\) with source \(L^r_{adm}\) instead. In the course of the next Proposition one can then shown that \(L^r_{adm} = L\).

  4. 4.

    For \(m>0\) one can extend the definition to \(\varprojlim _n L_n^\times \) while for \(m=0\) one cannot evaluate at \(\eta _0=0\)!

  5. 5.

    This power series has a constant term: see [16] for a technical solution.

  6. 6.

    It follows from [12, Proposition III.3.1] that this sequence splits in the category of topological \(\mathbf {Q}_p\)-vector spaces. Since the p-adic topology on \(\mathbf {Q}_p\) coincides with the induced topology from \(B_{max,\mathbf {Q}_p}\) the existence of the transition map is granted by [30, Lem. 2.7.2].

  7. 7.

    Analogous arguments as in Footnote 6 grant the existence of this connecting homomorphism.

  8. 8.

    Using the facts from Footnote 6 one checks that this sequence again satisfies the conditions of [30, Lem. 2.7.2] whence the existence of the long exact cohomology sequence below is granted.

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Authors and Affiliations

  1. Mathematisches Institut, Universität Münster, Einsteinstr. 62, 48291, Münster, Germany

    Peter Schneider

  2. Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 288, 69120, Heidelberg, Germany

    Otmar Venjakob

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  1. Mathematics Institute, University of Warwick, Coventry, United Kingdom

    David Loeffler

  2. Department of Mathematics, University College London, London, United Kingdom

    Sarah Livia Zerbes

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Schneider, P., Venjakob, O. (2016). Coates–Wiles Homomorphisms and Iwasawa Cohomology for Lubin–Tate Extensions. In: Loeffler, D., Zerbes, S. (eds) Elliptic Curves, Modular Forms and Iwasawa Theory. JHC70 2015. Springer Proceedings in Mathematics & Statistics, vol 188. Springer, Cham. https://doi.org/10.1007/978-3-319-45032-2_12

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