Witte, Malte

Non-Commutative \(L\)-Functions for \(p\)-Adic Representations over Totally Real Fields

Doc. Math. 24, 1413-1511 (2019)
DOI: 10.25537/dm.2019v24.1413-1511
Communicated by Otmar Venjakob


We prove a unicity result for the non-commutative \(L\)-functions for \(p\)-adic representations over totally real fields-functions appearing in the non-commutative Iwasawa main conjecture over totally real fields. We then consider continuous representations \(\rho\) of the absolute Galois group of a totally real field \(F\) on adic rings in the sense of Fukaya and Kato. Using our unicity result, we show that there exists a unique sensible definition of a non-commutative \(L\)-function for any such \(\rho\) that factors through the Galois group of a possibly infinite totally real extension. We also consider the case of CM-extensions and discuss the relation with the equivariant main conjecture for realisations of abstract 1-motives of Greither and Popescu.

Mathematics Subject Classification

11R23, 11R42, 19F27


main conjecture, non-commutative Iwasawa theory, totally real fields


  • [AGV72a]. M. Artin, A. Grothendieck, and J.L. Verdier, Théorie des topos et cohomologie étale des schémas (SGA 4-2), Lecture Notes in Mathematics, no. 270, Springer, Berlin, 1972. DOI 10.1007/BFb0061319; zbl 0237.00012.
  • [AGV72b]. M. Artin, A. Grothendieck, and J.L. Verdier, Théorie des topos et cohomologie étale des schémas (SGA 4-3), Lecture Notes in Mathematics, no. 305, Springer, Berlin, 1972. DOI 10.1007/BFb0070714; zbl 0245.00002.
  • [Bau91]. H. J. Baues, Combinatorial homotopy and \(4\)-dimensional complexes, de Gruyter Expositions in Mathematics, no. 2, Walter de Gruyter \& Co., Berlin, 1991. zbl 0716.55001; MR1096295.
  • [Bur09]. D. Burns, Algebraic \(p\)-adic \(L\)-functions in non-commutative Iwasawa theory, Publ. RIMS Kyoto 45 (2009), 75-88. DOI 10.2977/prims/1234361155; zbl 1225.11137; MR2512778.
  • [Bur15]. D. Burns, On main conjectures in non-commutative Iwasawa theory and related conjectures, J. Reine Angew. Math. 698 (2015), 105-159. DOI 10.1515/crelle-2013-0001; zbl 1322.11110; MR3294653.
  • [CFK{\etalchar{+}}05]. J. Coates, T. Fukaya, K. Kato, R. Sujatha, and O. Venjakob, The \(GL_2\) main conjecture for elliptic curves without complex multiplication, Publ. Math. Inst. Hautes Etudes Sci. (2005), no. 101, 163-208. DOI 10.1007/s10240-004-0029-3; zbl 1108.11081; MR2217048; arxiv math/0404297.
  • [CL73]. J. Coates and S. Lichtenbaum, On \(l\)-adic zeta functions, Ann. of Math. (2) 98 (1973), 498-550. DOI 10.2307/1970916; zbl 0279.12005; MR0330107.
  • [CSSV13]. J. Coates, P. Schneider, R. Sujatha, and O. Venjakob (eds.), Noncommutative Iwasawa main conjectures over totally real fields, Springer Proceedings in Mathematics \& Statistics, vol. 29, Springer, Heidelberg, 2013, Papers from the Workshop held at the University of Münster, Münster, April 25-30, 2011. DOI 10.1007/s00605-016-0901-5; zbl 1336.00009; MR3068893.
  • [Del87]. P. Deligne, Le déterminant de la cohomologie, Contemporary Mathematics 67 (1987), 93-177. zbl 0629.14008; MR0902592.
  • [FK06]. T. Fukaya and K. Kato, A formulation of conjectures on \(p\)-adic zeta functions in non-commutative Iwasawa theory, Proceedings of the St. Petersburg Mathematical Society (Providence, RI), vol. XII, Amer. Math. Soc. Transl. Ser. 2, no. 219, American Math. Soc., 2006, pp. 1-85. zbl 1238.11105; MR2276851.
  • [Fu11]. L. Fu, Etale cohomology theory, Nankai Tracts in Mathematics, vol. 13, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. zbl 1228.14001; MR2791606.
  • [GP15]. C. Greither and C. Popescu, An equivariant main conjecture in Iwasawa theory and applications, J. Algebraic Geom. 24 (2015), no. 4, 629-692. DOI 10.1090/jag/635; zbl 1330.11070; MR3383600; arxiv 1103.3069.
  • [GP17]. C. Greither and C. Popescu, Abstract \(\ell \)-adic \(1\)-motives and Tate's canonical class for number fields, Doc. Math. 23 (2018), 839-870. https://www.elibm.org/article/10011862; DOI 10.25537/dm.2018v23.839-870; zbl 1405.11144; MR3861043; arxiv 1710.02596.
  • [Gre83]. R. Greenberg, On \(p\)-adic Artin \(L\)-functions, Nagoya Math. J. 89 (1983), 77-87. DOI 10.1017/S0027763000020250; zbl 0513.12012; MR0692344.
  • [Gro60]. A. Grothendieck, Éléments de géométrie algébrique: I. Le langage des schémas, no. 4, Inst. Hautes Études Sci. Publ. Math., 1960. zbl 0118.36206 (3 volumes); MR0217083.
  • [GW04]. K. R. Goodearl and R. B. Warfield, An introduction to noncommutative noetherian rings, 2 ed., London Math. Soc. Student Texts, no. 61, Cambridge Univ. Press, Cambridge, 2004. zbl 1101.16001; MR2080008.
  • [Kak13]. M. Kakde, The main conjecture of Iwasawa theory for totally real fields, Invent. Math. 193 (2013), no. 3, 539-626. DOI 10.1007/s00222-012-0436-x; zbl 1300.11112; MR3091976; arxiv 1008.0142.
  • [Kat06]. K. Kato, Iwasawa theory for totally real fields for Galois extensions of Heisenberg type, preprint, 2006.
  • [KW01]. R. Kiehl and R. Weissauer, Weil conjectures, perverse sheaves and \(l\)'adic Fourier transform, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 42, Springer-Verlag, Berlin, 2001. zbl 0988.14009; MR1855066.
  • [Mih16]. P. Mihuailescu, On the vanishing of Iwasawa's constant \(\mu\) for the cyclotomic \(\mathbb{Z}_p\)-extensions of CM number fields, February 2016. arxiv 1409.3114.
  • [Mil80]. J. S. Milne, Etale cohomology, Princeton Mathematical Series, no. 33, Princeton University Press, New Jersey, 1980. zbl 0433.14012; MR0559531.
  • [Mil06]. J. S. Milne, Arithmetic duality theorems, second ed., BookSurge, LLC, Charleston, SC, 2006. zbl 1127.14001; MR2261462.
  • [MT07]. F. Muro and A. Tonks, The 1-type of a Waldhausen K-theory spectrum, Advances in Mathematics 216 (2007), no. 1, 178-211. DOI 10.1016/j.aim.2007.05.008; zbl 1125.19001; MR2353254; arxiv math/0603544.
  • [MT08]. F. Muro and A. Tonks, On \(K_1\) of a Waldhausen category, K-Theory and Noncommutative Geometry, EMS Series of Congress Reports, 2008, pp. 91-116. zbl 1165.18010; MR2513334; arxiv math/0611229.
  • [MTW15]. Fernando Muro, Andrew Tonks, and Malte Witte, On determinant functors and \(K\)-theory, Publ. Mat. 59 (2015), no. 1, 137-233. DOI 10.5565/PUBLMAT_59115_07; zbl 1316.19002; MR3302579; arxiv 1006.5399.
  • [Nic13]. A. Nickel, Equivariant Iwasawa theory and non-abelian Stark-type conjectures, Proc. Lond. Math. Soc. (3) 106 (2013), no. 6, 1223-1247. DOI 10.1112/plms/pds086; zbl 1273.11155; MR3072281; arxiv abs/1006.5399.
  • [NSW00]. J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of number fields, Grundlehren der mathematischen Wissenschaften, no. 323, Springer Verlag, Berlin Heidelberg, 2000. zbl 0948.11001; MR1737196.
  • [Oli88]. R. Oliver, Whitehead groups of finite groups, London Mathematical Society lecture notes series, no. 132, Cambridge University Press, Cambridge, 1988. zbl 0636.18001; MR0933091.
  • [RW11]. J. Ritter and A. Weiss, On the ``main conjecture'' of equivariant Iwasawa theory, J. Amer. Math. Soc. 24 (2011), no. 4, 1015-1050. DOI 10.1090/S0894-0347-2011-00704-2; zbl 1228.11165; MR2813337; arxiv 1004.2578.
  • [Sch79]. P. Schneider, Über gewisse Galoiscohomologiegruppen, Math. Z. 260 (1979), 181-205. DOI 10.1007/BF01214195; zbl 0421.12024; MR0544704.
  • [SV13]. P. Schneider and O. Venjakob, \(SK_1\) and Lie algebras, Math. Ann. 357 (2013), no. 4, 1455-1483. DOI 10.1007/s00208-013-0943-0; zbl 1288.19002; MR3124938; arxiv 1107.6008.
  • [TT90]. R. W. Thomason and T. Trobaugh, Higher algebraic \(K\)-theory of schemes and derived categories, The Grothendieck Festschrift, vol. III, Progr. Math., no. 88, Birkhäuser, 1990, pp. 247-435. zbl 0731.14001; MR1106918.
  • [Wal85]. F. Waldhausen, Algebraic \(K\)-theory of spaces, Algebraic and Geometric Topology (Berlin Heidelberg), Lecture Notes in Mathematics, no. 1126, Springer, 1985, pp. 318-419. zbl 0579.18006; MR0802796.
  • [War93]. S. Warner, Topological rings, North-Holland Mathematical Studies, no. 178, Elsevier, Amsterdam, 1993. zbl 0785.13008; MR1240057.
  • [Was97]. L. C. Washington, Introduction to cyclotomic fields, 2 ed., Graduate Texts in Mathematics, no. 83, Springer-Verlag, New York, 1997. zbl 0966.11047; MR1421575.
  • [Wit]. M. Witte, On \(\zeta \)-isomorphisms for totally real fields, in preparation.
  • [Wit08]. M. Witte, Noncommutative Iwasawa main conjectures for varieties over finite fields, Ph.D. thesis, Universität Leipzig, 2008. http://d-nb.info/995008124/34.
  • [Wit13a]. M. Witte, Noncommutative main conjectures of geometric Iwasawa theory, Noncommutative Iwasawa Main Conjectures over Totally Real Fields (Heidelberg), PROMS, no. 29, Springer, 2013, pp. 183-206. DOI 10.1007/978-3-642-32199-3_7; zbl 1270.11113; MR3068898; arxiv 1111.1645.
  • [Wit13b]. M. Witte, On a localisation sequence for the K-theory of skew power series rings, J. K-Theory 11 (2013), no. 1, 125-154. DOI 10.1017/is013001019jkt198; zbl 1270.16035; MR3034286; arxiv 1109.3423.
  • [Wit13c]. M. Witte, On a noncommutative Iwasawa main conjecture for function fields, preprint, 2013. arxiv 1710.09113.
  • [Wit14]. M. Witte, On a noncommutative Iwasawa main conjecture for varieties over finite fields, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 2, 289-325. DOI 10.4171/JEMS/434; zbl 1290.14016; MR3161285; arxiv 1004.2481.
  • [Wit16]. M. Witte, Unit \(L\)-functions for étale sheaves of modules over noncommutative rings, Journal de théorie des nombres de Bordeaux 28 (2016), no. 1, 89-113. DOI 10.5802/jtnb.930; zbl 1359.14023; MR3464613; arxiv 1309.2493.
  • [Wit17]. M. Witte, Non-commutative Iwasawa theory for global fields, Habilitationsschrift, Ruprecht-Karls-Universität Heidelberg, 2017.
  • [WY92]. C. Weibel and D. Yao, Localization for the \(K\)-theory of noncommutative rings, Algebraic \(K\)-Theory, Commutative Algebra, and Algebraic Geometry, Contemporary Mathematics, no. 126, AMS, 1992, pp. 219-230. zbl 0780.19007; MR1156514.


Witte, Malte
Ruprecht-Karls-Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, D-69120 Heidelberg, Germany