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Local Tamagawa numbers and the Bloch-Kato conjecture for the motives \(\mathbb Q(m)\) over an abelian field. (Les nombres de Tamagawa locaux et la conjecture de Bloch et Kato pour les motifs \(\mathbb Q(m)\) sur un corps abélien.) (French) Zbl 1125.11351

In this instructive and nicely written article, the authors present a proof of the Bloch-Katō conjecture for the motive \(Z(m)\) over an arbitrary absolutely abelian number field \(F\), where \(m\) is an arbitrary integer. The cases \(m=0,1\) are known, and are not discussed any further. The main ingredients are a result on algebraic \(K\)-groups (cohomological Lichtenbaum conjectures) and explicit calculations. For \(F\) the rationals, the result is already in the ground-breaking paper of S. Bloch and K. Kato [in: The Grothendieck Festschrift, Vol. I, Prog. Math. 86, 333–400 (1990; Zbl 0768.14001)]. A somewhat more general result (validity of the Bloch-Kato conjecture for Dirichlet motives) was recently proved by Huber and Kings by entirely different methods, and a yet stronger result (to wit: a result on equivariant Tamagawa numbers) was given by D. Burns and the reviewer [Invent. Math. 153, No. 2, 303–359 (2003; Zbl 1142.11076)], again by a different approach and in a rather different mathematical language.
Let us briefly explain the setting and the approach used by the authors of the present paper. Given a motive \(M\) defined over \(F\) (for example \(M=Z(m)\)), Bloch and Kato associated with it, under certain assumptions, a Tamagawa number \(\text{Tam}(M)\) which roughly speaking encompasses Euler factors on the one hand and regulators on the other hand. In fact, \(\text{Tam}(M)\) is the measure of a certain compact abelian group attached to \(M\), whose definition resembles the construction of the idéle class group; calculating this measure involves calculations at finite places and something like a regulator calculation at the infinite places.
A little more precisely: In the present article, one deals with a global number \(\text{Tam}^0(M)\) which is the product of local contributions \(\text{Tam}^0_v(M)\), \(v\) running over the finite places of \(F\) and \(\infty\). In the Bloch-Katō article [loc. cit.], only the global Tamagawa number \(\text{Tam}(M)\) is explicitly defined, but one implicitly has local contributions \(\text{Tam}_v(M)\), denoted \(\mu_p(A(Q_p))\) in [S. J. Bloch and K. Katō, loc. cit.] for \(v\) finite (they assume \(F=Q\), so \(v\) corresponds to a prime \(p\)); these differ from \(\text{Tam}^0_v(M)\) by an Euler factor; cf. p. 240 in J.-M. Fontaine’s article [Astérisque No. 206, Exp. No. 751, 205–249 (1992; Zbl 0799.14006)]. Hence the conjecture C\(_{\text{BK}}\) involving \(\text{Tam}^0(M)\) as formulated in the article under review shows a zeta function on its left hand side, whereas the corresponding statement involving \(\text{Tam}(M)\) due to Bloch and Katō [loc. cit. (Section 5.15)] contains no zeta or \(L\)-function.
The authors prepare everything that is needed for the proof of C\(_{\text{BK}}\) for \(Z(m)\) in Sections 2 to 4: they calculate the local Tamagawa numbers outside infinity (this generalizes the corresponding result for the unramified case in the Bloch-Kato paper [loc. cit.], they determine the Tamagawa number at infinity, using the Beilinson regulator, and also the Ш groups, which are closely related to higher \(K\)-groups. When all these calculations are finally assembled, to obtain the desired result it suffices to invoke the validity of the (cohomological) Lichtenbaum conjecture which relates the leading term of the zeta function at \(s=1-m\) (for \(m\geq2\)), the Beilinson regulator, and the size of the \((2m-2)\)nd cohomological \(K\)-group. This validity was proved, for the absolutely abelian case, by M. Kolster, T. Nguyen Quang Do and V. Fleckinger [Duke Math. J. 84, 679–717 (1996; Zbl 0863.19003)]. (Remark: In an appendix, the authors of the present article take the opportunity of carefully cleaning up a few technical inaccuracies in that work; this mainly concerns superfluous Euler factors and one inappropriate reference.) In the final part of the proof, there is a case distinction \(m\leq-1\) and \(m\geq 2\). In the proof of the latter case, the functional equation for the zeta function comes in. The authors remark at an early stage that their results imply the compatibility of the Bloch-Katō conjecture with the functional equation.
The calculation of the Tamagawa numbers at finite places for the motives \(Z(m)\) involves finding the order of the cokernel of the Bloch-Kato exponential. This is done using a result of Perrin-Riou generalizing Coleman theory, and the calculation of the relative index of two explicit lattices in \(F\).
In Section 2.4 the authors give some formulas for equivariant Tamagawa numbers, mentioning that these should suffice to get compatibility with the functional equation in the equivariant setting as well. It appears that Benois and Burns (work in progress) have results in this direction.

MSC:

11R42 Zeta functions and \(L\)-functions of number fields
11G55 Polylogarithms and relations with \(K\)-theory

References:

[1] Banaszak G , Generalization of the Moore exact sequence and the wild kernel for higher K -groups , Compositio Math. 86 ( 3 ) ( 1993 ) 281 - 305 . Numdam | MR 1219629 | Zbl 0778.11066 · Zbl 0778.11066
[2] Beilinson A., Polylogarithms and cyclotomic elements , Preprint, 1990.
[3] Belliard J.-R , Nguyen Quang Do T , Formules de classes pour les corps abéliens réels , Ann. Inst. Fourier 51 ( 4 ) ( 2001 ) 903 - 937 . Numdam | MR 1849210 | Zbl 1007.11063 · Zbl 1007.11063 · doi:10.5802/aif.1840
[4] Belliard J.-R., Nguyen Quang Do T., Modified circular p -units and annihilation of real classes , prépublication, 2001.
[5] Benois D., Burns D., travail en préparation.
[6] Borel A , Cohomologie de SL n et valeurs de fonctions zêta , Ann. Scuola Norm. Sup. Pisa 417 ( 1974 ) 613 - 636 . Numdam | MR 506168 | Zbl 0382.57027 · Zbl 0382.57027
[7] Bloch S , Kato K , L -functions and Tamagawa numbers of motives , Grothendieck Festschrift 1 ( 1990 ) 333 - 400 . MR 1086888 | Zbl 0768.14001 · Zbl 0768.14001
[8] Burns D , Flach M , Motivic L -functions and Galois module structure , Math. Ann. 305 ( 1996 ) 65 - 102 . MR 1386106 | Zbl 0867.11081 · Zbl 0867.11081 · doi:10.1007/BF01444212
[9] Burns D., Greither C., On the equivariant Tamagawa number conjecture for Tate motives , Preprint, 2000. MR 1992015 · Zbl 1142.11076
[10] Beilinson A , MacPherson R , Schechtman V , Notes on motivic cohomology , Duke Math. J. 54 ( 1987 ) 679 - 710 . Article | MR 899412 | Zbl 0632.14010 · Zbl 0632.14010 · doi:10.1215/S0012-7094-87-05430-5
[11] Coleman R , Local units modulo circular units , Proc. Amer. Math. Soc. 89 ( 1983 ) 1 - 7 . MR 706497 | Zbl 0528.12005 · Zbl 0528.12005 · doi:10.2307/2045050
[12] Deligne P , Le groupe fondamental de la droite projective moins trois points , in: Galois Groups Over Q , MSRI Publications , 16 , Springer , 1989 , pp. 79 - 297 . MR 1012168 | Zbl 0742.14022 · Zbl 0742.14022
[13] Dwyer W.-G , Friedlander E.M , Algebraic and etale K -theory , Trans. Amer. Math. Soc. 292 ( 1985 ) 247 - 280 . MR 805962 | Zbl 0581.14012 · Zbl 0581.14012 · doi:10.2307/2000179
[14] Flach M , A generalization of the Cassels-Tate pairing , J. Reine Angew. Math. 412 ( 1990 ) 113 - 127 . Article | MR 1079004 | Zbl 0711.14001 · Zbl 0711.14001 · doi:10.1515/crll.1990.412.113
[15] Fontaine J.-M , Sur certains types de représentations p -adiques du groupe de Galois d’un corps local ; construction d’un anneau de Barsotti-Tate , Ann. of Math. 115 ( 1982 ) 529 - 577 . MR 657238 | Zbl 0544.14016 · Zbl 0544.14016 · doi:10.2307/2007012
[16] Fontaine J.-M , Le corps des périodes p -adiques , Astérisque 223 ( 1994 ) 59 - 102 . MR 1293971
[17] Fontaine J.-M , Valeurs spéciales de fonctions L des motifs , Séminaire Bourbaki, exposé 751, Astérisque 206 ( 1992 ) 205 - 249 . Numdam | MR 1206069 | Zbl 0799.14006 · Zbl 0799.14006
[18] Fontaine J.-M , Perrin-Riou B , Autour des conjectures de Bloch et Kato ; cohomologie galoisienne et valeurs de fonctions L , in: Motives , Proc. Symp. in Pure Math. , 55 , 1994 , pp. 599 - 706 . MR 1265546 | Zbl 0821.14013 · Zbl 0821.14013
[19] Gillard R , Unité cyclotomiques, unités semi-locales et Z l -extensions , Ann. Inst. Fourier 29 ( 1 ) ( 1979 ) 49 - 79 . Numdam | MR 526777 | Zbl 0387.12002 · Zbl 0387.12002 · doi:10.5802/aif.727
[20] Greither C , Class groups of abelian fields and the main conjecture , Ann. Inst. Fourier 42 ( 1992 ) 449 - 499 . Numdam | MR 1182638 | Zbl 0729.11053 · Zbl 0729.11053 · doi:10.5802/aif.1299
[21] Gross B.H., On the values of Artin L -functions , Preprint, 1980. MR 2154331
[22] Huber A., Kings G., Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters , Preprint, 2000. arXiv | MR 2002643 · Zbl 1044.11095
[23] Huber A , Wildeshaus J , Classical motivic polylogarithm according to Beilinson and Deligne , Doc. Math. J. DMV3 ( 1998 ) 27 - 133 . MR 1643974 | Zbl 0906.19004 · Zbl 0906.19004
[24] Kahn B , On the Lichtenbaum-Quillen conjecture , in: Algebraic K -theory and Algebraic Topology , NATO Proc. Lake Louise , 407 , 1993 , pp. 147 - 166 . MR 1367295 | Zbl 0885.19004 · Zbl 0885.19004
[25] Kato K , Lectures on the approach to Iwasawa theory for Hasse-Weil L -functions via B dR . Part I , in: Lecture Notes in Math. , 1553 , Springer , 1993 , pp. 50 - 163 . MR 1338860 | Zbl 0815.11051 · Zbl 0815.11051
[26] Kato K., Lectures on the approach to Iwasawa theory for Hasse-Weil L -functions via B dR . Part II , Preprint, 1993. MR 1338860 · Zbl 0815.11051
[27] Kolster M., Nguyen Quang Do T., Universal distribution lattices for abelian number fields , Preprint, 2000.
[28] Kolster M , Nguyen Quang Do T , Fleckinger V , Twisted S -units, p -adic class number formulas and the Lichtenbaum conjectures , Duke Math. J. 84 ( 1996 ) 679 - 717 . Article | MR 1408541 | Zbl 0863.19003 · Zbl 0863.19003 · doi:10.1215/S0012-7094-96-08421-5
[29] Kuzmin L.V , On formulae for the class number of real abelian fields , Russian Math. Izv. 60 ( 4 ) ( 1996 ) 695 - 761 . MR 1416925 | Zbl 1007.11065 · Zbl 1007.11065 · doi:10.1070/IM1996v060n04ABEH000079
[30] Leopoldt H.-W , Über die Hauptordnung der ganzen Elemente eines abelschen Zahlkörpers , J. Reine Angew. Math. 201 ( 1959 ) 119 - 149 . Article | MR 108479 | Zbl 0098.03403 · Zbl 0098.03403 · doi:10.1515/crll.1959.201.119
[31] Lettl G , Relative Galois module structure of integers of local abelian fields , Acta Arithmetica 85 ( 3 ) ( 1998 ) 235 - 247 . Article | MR 1627831 | Zbl 0910.11050 · Zbl 0910.11050
[32] Milne J.S , Arithmetic Duality Theorems , Perspectives in Mathematics , 1 , Academic Press , Boston , 1986 . MR 881804 | Zbl 0613.14019 · Zbl 0613.14019
[33] Neukirch J , The Beilinson conjecture for algebraic number fields , in: Beilinson’s Conjectures on Special Values of L -functions , Perspectives in Math. , 4 , Academic Press , 1988 , pp. 193 - 247 . MR 944995 | Zbl 0651.12009 · Zbl 0651.12009
[34] Nguyen Quang Do T , Analogues supérieurs du noyau sauvage , J. Théorie des Nombres Bordeaux 4 ( 1992 ) 263 - 271 . Numdam | MR 1208865 | Zbl 0783.11042 · Zbl 0783.11042 · doi:10.5802/jtnb.74
[35] Perrin-Riou B , Théorie d’Iwasawa des représentations p -adiques sur un corps local , Invent. Math. 115 ( 1994 ) 81 - 149 . MR 1248080 | Zbl 0838.11071 · Zbl 0838.11071 · doi:10.1007/BF01231755
[36] Perrin-Riou B , Fonctions L p -adiques , in: Proc. Int. Congress of Math. , Birkhäuser Verlag , Zürich , 1995 , pp. 400 - 410 . MR 1403940 | Zbl 0853.11093 · Zbl 0853.11093
[37] Perrin-Riou B , Systèmes d’Euler p -adiques et théorie d’Iwasawa , Annales de l’Institut Fourier 48 ( 5 ) ( 1998 ) 1231 - 1307 . Numdam | MR 1662231 | Zbl 0930.11078 · Zbl 0930.11078 · doi:10.5802/aif.1655
[38] Schneider P , Über gewisse Galoiscohomologiegruppen , Math. Zeit. 168 ( 1979 ) 181 - 205 . Article | MR 544704 | Zbl 0421.12024 · Zbl 0421.12024 · doi:10.1007/BF01214195
[39] Schneider P , Introduction to the Beilinson conjectures , in: Beilinson’s Conjectures on Special Values of L -functions , Perspectives in Math. , 4 , Academic Press , 1988 , pp. 1 - 35 . MR 944989 | Zbl 0673.14007 · Zbl 0673.14007
[40] Sinnott W , On the Stickelberger ideal and the circular units of an abelian field , Invent. Math. 62 ( 1981 ) 181 - 234 . MR 595586 | Zbl 0465.12001 · Zbl 0465.12001 · doi:10.1007/BF01389158
[41] Solomon D , On a construction of p -units in abelian fields , Invent. Math. 109 ( 1992 ) 329 - 350 . MR 1172694 | Zbl 0772.11043 · Zbl 0772.11043 · doi:10.1007/BF01232030
[42] Soulé C , K -théorie des anneaux d’entiers de corps de nombres et cohomologie étale , Invent. Math. 55 ( 1979 ) 251 - 295 . MR 553999 | Zbl 0437.12008 · Zbl 0437.12008 · doi:10.1007/BF01406843
[43] Soulé C , Régulateurs , Sém. Bourbaki (1984/85), exp. n^\circ 644, Astérisque 133-134 ( 1986 ) 237 - 253 . Numdam | MR 837223 | Zbl 0617.14008 · Zbl 0617.14008
[44] Tate J , Relations between K 2 and Galois cohomology , Invent. Math. 36 ( 1976 ) 257 - 274 . MR 429837 | Zbl 0359.12011 · Zbl 0359.12011 · doi:10.1007/BF01390012
[45] Tsuji T , Semi-local units modulo cyclotomic units , J. Number Theory 46 ( 1999 ) 158 - 178 . MR 1706941 | Zbl 0948.11042 · Zbl 0948.11042 · doi:10.1006/jnth.1999.2398
[46] Villemot L., Étude du quotient des unités semi-locales par les unités cyclotomiques dans les Z p -extensions des corps de nombres abéliens réels , thèse, Orsay, 1981. MR 627614 | Zbl 0473.12003 · Zbl 0473.12003
[47] Washington L.C , Introduction to the Theory of Cyclotomic Fields , GTM , 85 , Springer , 1982 . MR 718674 · Zbl 0484.12001
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