×

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
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] D. Benois, On the Iwasawa theory of crystalline representations , Duke Math. J. 104 (2000), 211-267. · Zbl 0996.11072 · doi:10.1215/S0012-7094-00-10422-X
[2] D. Benois and L. Berger, Théorie d’Iwasawa des représentations cristallines, II , Comment. Math. Helv. 83 (2008), 603-677. · Zbl 1157.11041 · doi:10.4171/CMH/138
[3] L. Berger, Bloch and Kato’s exponential map: Three explicit formulas , Doc. Math. Extra Vol. (2003), 99-129. · Zbl 1064.11077
[4] Berger, L., Limites de représentations cristallines , Compos. Math. 140 (2004), 1473-1498. · Zbl 1071.11067 · doi:10.1112/S0010437X04000879
[5] S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean Analysis: A Systematic Approach to Rigid Analytic Geometry , Grundlehren Math. Wiss. 261 , Springer, Berlin, 1984. · Zbl 0539.14017
[6] M. Breuning, Determinant functors on triangulated categories , J. K-Theory 8 (2011), 251-291. · Zbl 1243.18023 · doi:10.1017/is010006009jkt120
[7] J. Coates, T. Fukaya, K. Kato, R. Sujatha, and O. Venjakob, The \(\operatorname{GL}_{2}\) main conjecture for elliptic curves without complex multiplication , Publ. Math. Inst. Hautes Études Sci. 101 (2005), 163-208. · Zbl 1108.11081 · doi:10.1007/s10240-004-0029-3
[8] P. Colmez, Théorie d’Iwasawa des représentations de Rham d’un corps local , Ann. of Math. (2) 148 (1998), 485-571. · Zbl 0928.11045 · doi:10.2307/121003
[9] P. Deligne, “Les constantes des équations fonctionnelles des fonctions \(L\)” in Modular Functions of One Variable, II (Antwerp, 1972) , Lecture Notes in Math. 349 , Springer, Berlin, 1973, 501-597. · doi:10.1007/978-3-540-37855-6_7
[10] P. Deligne, “Le déterminant de la cohomologie” in Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985) , Contemp. Math. 67 , Amer. Math. Soc., Providence, 1987, 93-177. · doi:10.1090/conm/067/902592
[11] J.-M. Fontaine, “Représentations \(p\)-adiques semi-stables,” with an appendix by P. Colmez, in Périodes \(p\)-adiques (Bures-sur-Yvette, France, 1988) , Astérisque 223 , Soc. Math. France, Montrouge, 1994, 113-184.
[12] T. Fukaya and K. Kato, “A formulation of conjectures on \(p\)-adic zeta functions in noncommutative Iwasawa theory” in Proceedings of the St. Petersburg Mathematical Society. Vol. XII , Amer. Math. Soc. Transl. Ser. (2) 219 , Amer. Math. Soc., Providence, 2006, 1-85. · Zbl 1238.11105
[13] F. F. Knudsen, Determinant functors on exact categories and their extensions to categories of bounded complexes , Michigan Math. J. 50 (2002), 407-444. · Zbl 1023.18012 · doi:10.1307/mmj/1028575741
[14] M. Lazard, Les zéros des fonctions analytiques d’une variable sur un corps valué complet , Publ. Math. Inst. Hautes Études Sci. 14 (1962), 47-75. · Zbl 0119.03701 · doi:10.1007/BF02684326
[15] A. Lei, D. Loeffler, and S. L. Zerbes, Wach modules and Iwasawa theory for modular forms , Asian J. Math. 14 (2010), no. 4, 475-528. · Zbl 1281.11095 · doi:10.4310/AJM.2010.v14.n4.a2
[16] A. Lei, D. Loeffler, S. L. Zerbes, Coleman maps and the \(p\)-adic regulator , Algebra Number Theory 5 (2011), 1095-1131. · Zbl 1271.11100 · doi:10.2140/ant.2011.5.1095
[17] D. Loeffler and S. L. Zerbes, Iwasawa theory and \(p\)-adic \(L\)-functions for \(\mathbf{Z}_{p}^{2}\)-extensions , Int. J. Number Theory 10 (2014), 2045-2095. · Zbl 1314.11066 · doi:10.1142/S1793042114500699
[18] K. Nakamura, A generalization of Kato’s local epsilon-conjecture for (phi, Gamma)-modules over the Robba ring , preprint, [math.NT] arXiv:1305.0880v2 · Zbl 1431.11072
[19] J. Nekovář, Selmer Complexes , Astérisque 310 , Soc. Math. France, Montrouge, 2006, viii+559.
[20] B. Perrin-Riou, Théorie d’Iwasawa et hauteurs \(p\)-adiques , Invent. Math. 109 (1992), 137-185. · Zbl 0781.14013 · doi:10.1007/BF01232022
[21] B. Perrin-Riou, Théorie d’Iwasawa des représentations \(p\)-adiques sur un corps local , with an appendix by J.-M. Fontaine, Invent. Math. 115 (1994), 81-161. · Zbl 0838.11071 · doi:10.1007/BF01231755
[22] B. Perrin-Riou, Théorie d’Iwasawa et loi explicite de réciprocité , Doc. Math. 4 (1999), 219-273.
[23] P. Schneider and J. Teitelbaum, Algebras of \(p\)-adic distributions and admissible representations , Invent. Math. 153 (2003), 145-196. · Zbl 1028.11070 · doi:10.1007/s00222-002-0284-1
[24] O. Venjakob, On Kato’s local \(\varepsilon\)-isomorphism conjecture for rank one Iwasawa modules , Algebra Number Theory 7 (2013), 2369-2416. · Zbl 1305.11095 · doi:10.2140/ant.2013.7.2369
[25] O. Venjakob, A note on determinant functors and spectral sequences , preprint, 2012, available at (accessed 23 January 2015).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.