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Local epsilon isomorphisms. (English) Zbl 1322.11112

In this paper the authors prove the so-called local \(\varepsilon\)-isomorphism conjecture formulated by T. Fukaya and K. Kato [Proc. St. Petersb. Math. Soc. 12. Transl., Ser. 2, Am. Math. Soc. 219, 1–85 (2006; Zbl 1238.11105)] in certain special cases. This generalizes work of D. Benois and L. Berger [Comment. Math. Helv. 83, No. 3, 603–677 (2008; Zbl 1157.11041)].
Let \(p\) be an odd prime and let \(L\) be a complete discretely valued extension of \(\mathbb Q_p\) with ring of integers \(\mathcal O\). Let \(T\) be an \(\mathcal O\)-lattice in a crystalline \(L\)-linear representation \(V\). Finally, let \(F\) be a finite unramified extension of \(\mathbb Q_p\), \(F_{\infty}\) the unique unramified \(\mathbb Z_p\)-extension of \(F\) and \(K_{\infty} := F_{\infty}(\mu_{p^{\infty}})\). Then \(G := \mathrm{Gal}(K_{\infty} / \mathbb Q_p)\) is a \(p\)-adic Lie extension of dimension two, and the Iwasawa cohomology complex \(R\Gamma_{Iw}(K_{\infty}, T)\) is a perfect complex of \(\Lambda_{\mathcal O}(G)\)-modules, where as usual \(\Lambda_{\mathcal O}(G)\) denotes the Iwasawa algebra of \(G\) with coefficients in \(\mathcal O\). Let \(\tilde{\mathcal O}\) be the ring of integers of \(\tilde L\), the tensor product (over \(\mathbb Q_p\)) of \(L\) with the completion of the maximal unramified extension of \(\mathbb Q_p\). The authors construct a canonical isomorphism of determinants over \(\Lambda_{\tilde{\mathcal O}}(G)\) \[ \begin{split} \varepsilon_{\Lambda_{\tilde{\mathcal O}}(G)}(T): \mathrm{Det}_{\Lambda_{\tilde{\mathcal O}}(G)}(0) \simeq \\ \Lambda_{\tilde{\mathcal O}}(G) \otimes \left(\mathrm{Det}_{\Lambda_{\mathcal O}(G)} (R\Gamma_{Iw}(K_{\infty}, T)) \cdot \mathrm{Det}_{\Lambda_{\mathcal O}(G)} (\Lambda_{\mathcal O}(G) \otimes_{\mathcal O} \mathbb D_{\mathrm{cris}}(T)) \right), \end{split} \] where \(\mathbb D_{\mathrm{cris}}(T)\) is a certain \(\mathcal O\)-lattice in \(\mathbb D_{\mathrm{cris}}(V)\). After base change to the total ring of quotients of the distribution algebra \(\mathcal H_{\tilde L}(G)\), this isomorphism is simply given by the two-variable \(p\)-adic regulator map of the first and the third author [Int. J. Number Theory 10, No. 8, 2045–2095 (2014; Zbl 1314.11066)] (this uses the fact that all Iwasawa cohomology groups \(H^i_{Iw}(K_{\infty}, V)\) are torsion unless \(i=1\)). To obtain an isomorphism on integral level, one has to divide out by certain correction factors purely determined by the Hodge-Tate weights of \(V\). This might be seen as a “local Iwasawa main conjecture”.
Via base change, one obtains \(\varepsilon\)-isomorphisms \(\varepsilon_R(T)\) for a more general class of rings \(R\). It is shown that these \(\varepsilon\)-isomorphisms satisfy many of the conditions conjectured by Fukaya and Kato. This includes, for instance, the behaviour under short exact sequences of lattices \(T\), base change, and local duality. The last section then deals with the compatibility of the above \(\varepsilon\)-isomorphism with the “standard” \(\varepsilon\)-isomorphism for \(V\) and all twists of \(V\) by de Rham characters of \(G\).

MSC:

11R23 Iwasawa theory
11S40 Zeta functions and \(L\)-functions

References:

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