×

Coates-Wiles homomorphisms and Iwasawa cohomology for Lubin-Tate extensions. (English) Zbl 1409.11107

Loeffler, David (ed.) et al., Elliptic curves, modular forms and Iwasawa theory. In honour of John H. Coates’ 70th birthday, Cambridge, UK, March 2015. Proceedings of the conference and the workshop. Cham: Springer. Springer Proc. Math. Stat. 188, 401-468 (2016).
Summary: 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 [L. Herr, Bull. Soc. Math. Fr. 126, No. 4, 563–600 (1998; Zbl 0967.11050); A. J. Scholl, Doc. Math. Extra Vol., 685–709 (2007; Zbl 1186.11070)]. 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 S. Bloch and K. Kato [Prog. Math. 86, 333–400 (1990; Zbl 0768.14001)] 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 \(T\).
For the entire collection see [Zbl 1364.11005].

MSC:

11S31 Class field theory; \(p\)-adic formal groups
11S25 Galois cohomology
11S37 Langlands-Weil conjectures, nonabelian class field theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Benois, D.: On Iwasawa theory of crystalline representations. Duke Math. J. 104, 211–267 (2000) · Zbl 0996.11072 · doi:10.1215/S0012-7094-00-10422-X
[2] Bentzen, S., Madsen, I.: Trace maps in algebraic K-theory and the Coates-Wiles homomorphism. J. Reine Angew. Math. 411, 171–195 (1990) · Zbl 0716.11055
[3] Bloch, S., Kato, K.: \[ L \] -functions and Tamagawa numbers of motives. In: The Grothendieck Festschrift, vol. I, 333–400, Progress Math., 86, Birkhäuser Boston (1990) · Zbl 0768.14001
[4] Bourbaki, N.: Topologie Générale. Chaps. 1–10. Springer (2007)
[5] Cherbonnier, F., Colmez, P.: Théorie d’Iwasawa des représentations \[ p \] -adiques d’un corps local. J. AMS 12, 241–268 (1999) · Zbl 0933.11056
[6] Coates J., Sujatha, R.: Cyclotomic fields and zeta values. Springer (2006) · Zbl 1100.11002
[7] Coates, J., Wiles, A.: On \[ p \] -adic \[ L \] -functions and elliptic units. J. Austral. Math. Soc. Ser. A 26(1), 1–25 (1978) · Zbl 0442.12007 · doi:10.1017/S1446788700011459
[8] Coleman, R.: Division values in local fields. Invent. math. 53, 91–116 (1979) · Zbl 0429.12010 · doi:10.1007/BF01390028
[9] Colmez, P.: Espaces de Banach de dimension finie. J. Inst. Math. Jussieu 1, 331–439 (2002) · Zbl 1044.11102 · doi:10.1017/S1474748002000099
[10] Colmez, P.: Fontaine’s rings and \[ p \] -adic \[ L \] -functions. Lecture Notes at Tsinghua Univ. (2004)
[11] Colmez, P.: \[ (\varphi ,\Gamma ) \] -modules et représentations du mirabolique de \[ {{\mathrm {GL}}_2}(\mathbb{Q}_p) \] . In: Berger, L., Breuil, C., Colmez, P. (eds.) Représentations \[ p \] -adiques de groupes \[ p \] -adiques, vol. II. Astérisque 330, 61–153 (2010) · Zbl 1235.11107
[12] Colmez, P.: Théorie d’Iwasawa des représentations de de Rham d’un corps local. Ann. Math. 148(2), 485–571 (1998) · Zbl 0928.11045 · doi:10.2307/121003
[13] Colmez, P.: A generalization of Coleman’s isomorphism. In: Algebraic Number Theory and Related Topics (Kyoto, 1997). Srikaisekikenkysho kkyroku 1026, 110–112 (1998) · Zbl 1016.11537
[14] de Shalit, E.: The explicit reciprocity law of Bloch–Kato. Columbia University Number Theory Seminar (New York, 1992). Astérisque 228(4), 197–221 (1995) · Zbl 0832.14012
[15] Fontaine, J.-M.: Répresentations \[ p \] -adiques des corps locaux. In: The Grothendieck Festschrift, vol. II, 249–309, Birkhäuser (1990) · Zbl 0743.11066
[16] Fontaine, J.-M.: Appendice: Sur un théorème de Bloch et Kato (lettre à B. Perrin-Riou). Invent. Math. 115, 151–161 (1994) · Zbl 0802.14010 · doi:10.1007/BF01231756
[17] Fourquaux, L., Xie, B.: Triangulable \[ O_F \] -analytic \[ (\varphi _q,\Gamma ) \] -modules of rank \[ 2 \] . Algebra Number Theory 7(10), 2545–2592 (2013) · Zbl 1297.11145 · doi:10.2140/ant.2013.7.2545
[18] Fukaya, T., Kato, K.: A formulation of conjectures on \[ p \] -adic zeta functions in non-commutative Iwasawa theory. In: Proceedings of St. Petersburg Math. Soc., vol. XII, AMS Transl. Ser. 2, vol. 219, 1–86 (2006) · Zbl 1238.11105
[19] Hazewinkel, M.: Formal Groups and Applications. Academic Press (1978) · Zbl 0454.14020
[20] Herr, L.: Sur la cohomologie galoisienne des corps \[ p \] -adiques. Bull. Soc. Math. France 126, 563–600 (1998) · Zbl 0967.11050 · doi:10.24033/bsmf.2337
[21] Hewitt, E., Ross, K.: Abstract Harmonic Analysis, vol. I. Springer (1994) · Zbl 0837.43002 · doi:10.1007/978-1-4419-8638-2
[22] Jensen, C.U.: Les Foncteurs Dérivés de \[ \varprojlim \] et leurs Applications en Théorie des Modules. Springer Lect. Notes Math., vol. 254 (1972) · Zbl 0238.18007 · doi:10.1007/BFb0058395
[23] Kato, K.: Lectures on the approach to Iwasawa theory for Hasse-Weil \[ L \] -functions via \[ B_{\mathrm {dR}} \] . I. Arithmetic algebraic geometry (Trento, 1991), Springer. Lect. Notes Math. 1553, 50–163 (1993) · doi:10.1007/BFb0084729
[24] Kisin, M., Ren, W.: Galois representations and Lubin-Tate groups. Documenta Math. 14, 441–461 (2009) · Zbl 1246.11112
[25] Kölcze P.: Ein Analogon zum Hilbertsymbol für algebraische Funktionen und Witt-Vektoren solcher Funktionen. Diplomarbeit (Betreuer: J. Neukirch) Universität Regensburg (1990)
[26] Lang, S.: Cyclotomic Fields. Springer (1978) · Zbl 0395.12005 · doi:10.1007/978-1-4612-9945-5
[27] Laubie, F.: Extensions de Lie et groupes d’automorphismes de corps locaux. Compositio Math. 67, 165–189 (1988) · Zbl 0649.12012
[28] Lazard, M.: Groupes analytiques \[ p \] -adiques. Publ. Math. IHES 26, 389–603 (1965) · Zbl 0139.02302
[29] Michael, E.: Continuous selections II. Ann. Math. 64, 562–580 (1956) · Zbl 0073.17702 · doi:10.2307/1969603
[30] Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of Number Fields. 2nd edn. Springer (2008) · Zbl 1136.11001 · doi:10.1007/978-3-540-37889-1
[31] Schneider, P.: Galois representations and \[ (\varphi ,\Gamma ) \] -modules. Lecture Notes, Münster (2015). http://wwwmath.uni-muenster.de/u/schneider/publ/lectnotes/index.html · Zbl 1383.11001
[32] Schneider, P., Vigneras, M.-F.: A functor from smooth \[ o \] -torsion representations to \[ (\varphi ,\Gamma ) \] -modules. In: Arthur, Cogdell, ... (eds.) On Certain L-Functions. Clay Math. Proc., vol. 13, 525–601, AMS-CMI (2011)
[33] Scholl, A. J.: Higher fields of norms and \[ (\phi ,\Gamma ) \] -modules. Documenta Math. 2006, Extra Vol., pp. 685–709 (2006) · Zbl 1186.11070
[34] Serre, J.-P.: Abelian \[ l \] -Adic Representations and Elliptic Curves. Benjamin, W.A (1968)
[35] Serre, J.-P.: Cohomologie Galoisienne. Springer Lect. Notes Math., vol. 5 (1973) · doi:10.1007/978-3-662-21553-1
[36] Thomas, L.: Ramification groups in Artin-Schreier-Witt extensions. J. Théorie des Nombres de Bordeaux 17, 689–720 (2005) · Zbl 1207.11109 · doi:10.5802/jtnb.514
[37] Wiles, A.: Higher explicit reciprocity laws. Ann. Math. 107(2), 235–254 (1978) · Zbl 0378.12006 · doi:10.2307/1971143
[38] Witt, E.: Zyklische Körper und Algebren der Charakteristik \[ p \] vom Grad \[ p^n \] . Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik \[ p \] . J. Reine Angew. Math. 176, 126–140 (1936)
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.