On the equivariant Tamagawa number conjecture for Tate motives and unconditional annihilation results

Johnston, Henri; Nickel, Andreas

doi:10.1090/tran/6453

On the equivariant Tamagawa number conjecture for Tate motives and unconditional annihilation results
HTML articles powered by AMS MathViewer

by Henri Johnston and Andreas Nickel PDF

Trans. Amer. Math. Soc. 368 (2016), 6539-6574 Request permission

Abstract:

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$. Let $p$ be a prime and let $r \leq 0$ be an integer. By examining the structure of the $p$-adic group ring $\mathbb {Z}_{p}[G]$, we prove many new cases of the $p$-part of the equivariant Tamagawa number conjecture (ETNC) for the pair $(h^{0}(\mathrm {Spec}(L))(r),\mathbb {Z}[G])$. The same methods can also be applied to other conjectures concerning the vanishing of certain elements in relative algebraic $K$-groups. We then prove a conjecture of Burns concerning the annihilation of class groups as Galois modules for a large class of interesting extensions, including cases in which the full ETNC is not known. Similarly, we construct annihilators of higher dimensional algebraic $K$-groups of the ring of integers in $L$.

References

Daniel Barsky, Fonctions zeta $p$-adiques d’une classe de rayon des corps de nombres totalement réels, Groupe d’Étude d’Analyse Ultramétrique (5e année: 1977/78), Secrétariat Math., Paris, 1978, pp. Exp. No. 16, 23 (French). MR 525346
W. Bley and D. Burns, Equivariant epsilon constants, discriminants and étale cohomology, Proc. London Math. Soc. (3) 87 (2003), no. 3, 545–590. MR 2005875, DOI 10.1112/S0024611503014217
Manuel Breuning and David Burns, Leading terms of Artin $L$-functions at $s=0$ and $s=1$, Compos. Math. 143 (2007), no. 6, 1427–1464. MR 2371375, DOI 10.1112/S0010437X07002874
Werner Bley and Manuel Breuning, Exact algorithms for $p$-adic fields and epsilon constant conjectures, Illinois J. Math. 52 (2008), no. 3, 773–797. MR 2546007
Manuel Breuning and David Burns, On equivariant Dedekind zeta-functions at $s=1$, Doc. Math. Extra vol.: Andrei A. Suslin sixtieth birthday (2010), 119–146. MR 2804251
Werner Bley and Ruben Debeerst, Algorithmic proof of the epsilon constant conjecture, Math. Comp. 82 (2013), no. 284, 2363–2387. MR 3073206, DOI 10.1090/S0025-5718-2013-02691-9
D. Burns and M. Flach, Tamagawa numbers for motives with (non-commutative) coefficients, Doc. Math. 6 (2001), 501–570. MR 1884523
D. Burns and M. Flach, Tamagawa numbers for motives with (noncommutative) coefficients. II, Amer. J. Math. 125 (2003), no. 3, 475–512. MR 1981031
David Burns and Matthias Flach, On the equivariant Tamagawa number conjecture for Tate motives. II, Doc. Math. Extra Vol. (2006), 133–163. MR 2290586
D. Burns and C. Greither, On the equivariant Tamagawa number conjecture for Tate motives, Invent. Math. 153 (2003), no. 2, 303–359. MR 1992015, DOI 10.1007/s00222-003-0291-x
David Burns and Henri Johnston, A non-abelian Stickelberger theorem, Compos. Math. 147 (2011), no. 1, 35–55. MR 2771125, DOI 10.1112/S0010437X10004859
Spencer Bloch and Kazuya Kato, $L$-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 333–400. MR 1086888
W. Bley, The conjecture of Chinburg-Stark for abelian extensions of a quadratic imaginary field, Habilitationsschrift, Report No. 123 (1998), Augsburg.
W. Bley, Equivariant Tamagawa number conjecture for abelian extensions of a quadratic imaginary field, Doc. Math. 11 (2006), 73–118. MR 2226270
M. Breuning, Equivariant epsilon constants for Galois extensions of number fields and $p$-adic fields, Ph.D. thesis, King’s College London, 2004.
Manuel Breuning, Equivariant local epsilon constants and étale cohomology, J. London Math. Soc. (2) 70 (2004), no. 2, 289–306. MR 2078894, DOI 10.1112/S002461070400554X
Manuel Breuning, On equivariant global epsilon constants for certain dihedral extensions, Math. Comp. 73 (2004), no. 246, 881–898. MR 2031413, DOI 10.1090/S0025-5718-03-01605-3
Paul Buckingham, The equivalence of Rubin’s conjecture and the ETNC/LRNC for certain biquadratic extensions, Glasg. Math. J. 56 (2014), no. 2, 335–353. MR 3187901, DOI 10.1017/S0017089513000281
D. Burns, Equivariant Tamagawa numbers and Galois module theory. I, Compositio Math. 129 (2001), no. 2, 203–237. MR 1863302, DOI 10.1023/A:1014502826745
David Burns, On derivatives of Artin $L$-series, Invent. Math. 186 (2011), no. 2, 291–371. MR 2845620, DOI 10.1007/s00222-011-0320-0
David Burns, On derivatives of $p$-adic $L$-series at $s=0$, preprint, 2012.
David Burns, On main conjectures in non-commutative Iwasawa theory and related conjectures, J. Reine Angew. Math. 698 (2015), 105–159. MR 3294653, DOI 10.1515/crelle-2013-0001
Werner Bley and Stephen M. J. Wilson, Computations in relative algebraic $K$-groups, LMS J. Comput. Math. 12 (2009), 166–194. MR 2564571, DOI 10.1112/S1461157000001480
T. Chinburg, On the Galois structure of algebraic integers and $S$-units, Invent. Math. 74 (1983), no. 3, 321–349. MR 724009, DOI 10.1007/BF01394240
Ted Chinburg, The analytic theory of multiplicative Galois structure, Mem. Amer. Math. Soc. 77 (1989), no. 395, iv+158. MR 973358, DOI 10.1090/memo/0395
Pierrette Cassou-Noguès, Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta $p$-adiques, Invent. Math. 51 (1979), no. 1, 29–59 (French). MR 524276, DOI 10.1007/BF01389911
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. II, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. With applications to finite groups and orders; A Wiley-Interscience Publication. MR 892316
J. Coates and W. Sinnott, An analogue of Stickelberger’s theorem for the higher $K$-groups, Invent. Math. 24 (1974), 149–161. MR 369322, DOI 10.1007/BF01404303
F. R. DeMeyer and G. J. Janusz, Group rings which are Azumaya algebras, Trans. Amer. Math. Soc. 279 (1983), no. 1, 389–395. MR 704622, DOI 10.1090/S0002-9947-1983-0704622-4
Pierre Deligne and Kenneth A. Ribet, Values of abelian $L$-functions at negative integers over totally real fields, Invent. Math. 59 (1980), no. 3, 227–286. MR 579702, DOI 10.1007/BF01453237
M. R. Darafsheh and H. Sharifi, Frobenius $\Bbb Q$-groups, Arch. Math. (Basel) 83 (2004), no. 2, 102–105. MR 2104937, DOI 10.1007/s00013-004-1020-4
Matthias Flach, The equivariant Tamagawa number conjecture: a survey, Stark’s conjectures: recent work and new directions, Contemp. Math., vol. 358, Amer. Math. Soc., Providence, RI, 2004, pp. 79–125. With an appendix by C. Greither. MR 2088713, DOI 10.1090/conm/358/06537
Matthias Flach, On the cyclotomic main conjecture for the prime 2, J. Reine Angew. Math. 661 (2011), 1–36. MR 2863902, DOI 10.1515/CRELLE.2011.082
Jean-Marc Fontaine and Bernadette Perrin-Riou, Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions $L$, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 599–706 (French). MR 1265546
Bruce Ferrero and Lawrence C. Washington, The Iwasawa invariant $\mu _{p}$ vanishes for abelian number fields, Ann. of Math. (2) 109 (1979), no. 2, 377–395. MR 528968, DOI 10.2307/1971116
Roland Gillard, Fonctions $L$ $p$-adiques des corps quadratiques imaginaires et de leurs extensions abéliennes, J. Reine Angew. Math. 358 (1985), 76–91 (French). MR 797675, DOI 10.1515/crll.1985.358.76
K. W. Gruenberg, J. Ritter, and A. Weiss, A local approach to Chinburg’s root number conjecture, Proc. London Math. Soc. (3) 79 (1999), no. 1, 47–80. MR 1687551, DOI 10.1112/S0024611599011922
W. M. Hubbart, Some results on blocks over local fields, Pacific J. Math. 40 (1972), 101–109. MR 302752
H. Jacobinski, On extensions of lattices, Michigan Math. J. 13 (1966), 471–475. MR 204538
G. J. Janusz, Tensor products of orders, J. London Math. Soc. (2) 20 (1979), no. 2, 186–192. MR 551444, DOI 10.1112/jlms/s2-20.2.186
Jennifer Johnson-Leung, The local equivariant Tamagawa number conjecture for almost abelian extensions, Women in numbers 2: research directions in number theory, Contemp. Math., vol. 606, Amer. Math. Soc., Providence, RI, 2013, pp. 1–27. MR 3204289, DOI 10.1090/conm/606/12137
Jennifer Johnson-Leung and Guido Kings, On the equivariant main conjecture for imaginary quadratic fields, J. Reine Angew. Math. 653 (2011), 75–114. MR 2794626, DOI 10.1515/CRELLE.2011.020
Henri Johnston and Andreas Nickel, Noncommutative Fitting invariants and improved annihilation results, J. Lond. Math. Soc. (2) 88 (2013), no. 1, 137–160. MR 3092262, DOI 10.1112/jlms/jdt009
John W. Jones and David P. Roberts, A database of local fields, J. Symbolic Comput. 41 (2006), no. 1, 80–97. MR 2194887, DOI 10.1016/j.jsc.2005.09.003
Christian U. Jensen and Noriko Yui, Polynomials with $D_{p}$ as Galois group, J. Number Theory 15 (1982), no. 3, 347–375. MR 680538, DOI 10.1016/0022-314X(82)90038-5
Kazuya Kato, Lectures on the approach to Iwasawa theory for Hasse-Weil $L$-functions via $B_\textrm {dR}$. I, Arithmetic algebraic geometry (Trento, 1991) Lecture Notes in Math., vol. 1553, Springer, Berlin, 1993, pp. 50–163. MR 1338860, DOI 10.1007/BFb0084729
S. Kim, On the equivariant Tamagawa number conjecture for quaternion fields, Ph.D. thesis, King’s College London, 2001.
Burkhard Külshammer, Central idempotents in $p$-adic group rings, J. Austral. Math. Soc. Ser. A 56 (1994), no. 2, 278–289. MR 1261587
Tejaswi Navilarekallu, On the equivariant Tamagawa number conjecture for $A_4$-extensions of number fields, J. Number Theory 121 (2006), no. 1, 67–89. MR 2268756, DOI 10.1016/j.jnt.2006.01.003
A. Nickel, Integrality of Stickelberger elements and the equivariant Tamagawa number conjecture, to appear in Crelle, DOI 10.1515/crelle-2014-0042.
Andreas Nickel, Non-commutative Fitting invariants and annihilation of class groups, J. Algebra 323 (2010), no. 10, 2756–2778. MR 2609173, DOI 10.1016/j.jalgebra.2010.02.039
Andreas Nickel, Leading terms of Artin $L$-series at negative integers and annihilation of higher $K$-groups, Math. Proc. Cambridge Philos. Soc. 151 (2011), no. 1, 1–22. MR 2801311, DOI 10.1017/S0305004111000193
Andreas Nickel, On the equivariant Tamagawa number conjecture in tame CM-extensions, II, Compos. Math. 147 (2011), no. 4, 1179–1204. MR 2822866, DOI 10.1112/S0010437X11005331
Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2008. MR 2392026, DOI 10.1007/978-3-540-37889-1
H. Oukhaba and S. Viguié, On the $\mu$-invariant of Katz $p$-adic $L$-functions attached to imaginary quadratic fields and applications, http://arxiv.org/abs/1311.3565 (2013).
I. Reiner, Maximal orders, London Mathematical Society Monographs. New Series, vol. 28, The Clarendon Press, Oxford University Press, Oxford, 2003. Corrected reprint of the 1975 original; With a foreword by M. J. Taylor. MR 1972204
Derek J. S. Robinson, A course in the theory of groups, 2nd ed., Graduate Texts in Mathematics, vol. 80, Springer-Verlag, New York, 1996. MR 1357169, DOI 10.1007/978-1-4419-8594-1
Jürgen Ritter and Alfred Weiss, Cohomology of units and $L$-values at zero, J. Amer. Math. Soc. 10 (1997), no. 3, 513–552. MR 1423032, DOI 10.1090/S0894-0347-97-00234-8
Luis Ribes and Pavel Zalesskii, Profinite groups, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 40, Springer-Verlag, Berlin, 2010. MR 2599132, DOI 10.1007/978-3-642-01642-4
Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380
Victor P. Snaith, Stark’s conjecture and new Stickelberger phenomena, Canad. J. Math. 58 (2006), no. 2, 419–448. MR 2209286, DOI 10.4153/CJM-2006-018-5
C. Soulé, $K$-théorie des anneaux d’entiers de corps de nombres et cohomologie étale, Invent. Math. 55 (1979), no. 3, 251–295 (French). MR 553999, DOI 10.1007/BF01406843
R. G. Swan, Algebraic $K$-theory, Lecture Notes in Mathematics, No. 76, Springer-Verlag, Berlin-New York, 1968. MR 0245634
John Tate, Les conjectures de Stark sur les fonctions $L$ d’Artin en $s=0$, Progress in Mathematics, vol. 47, Birkhäuser Boston, Inc., Boston, MA, 1984 (French). Lecture notes edited by Dominique Bernardi and Norbert Schappacher. MR 782485
C. Weibel, The norm residue isomorphism theorem, J. Topol. 2 (2009), no. 2, 346–372. MR 2529300, DOI 10.1112/jtopol/jtp013
Da-Sheng Wei and Chun-Gang Ji, On the number of certain Galois extensions of local fields, Proc. Amer. Math. Soc. 135 (2007), no. 10, 3041–3047. MR 2322733, DOI 10.1090/S0002-9939-07-08905-8