Algebra & Number Theory

On Iwasawa theory, zeta elements for $\mathbb{G}_{m}$, and the equivariant Tamagawa number conjecture

David Burns, Masato Kurihara, and Takamichi Sano

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We develop an explicit “higher-rank” Iwasawa theory for zeta elements associated to the multiplicative group over abelian extensions of number fields. We show this theory leads to a concrete new strategy for proving special cases of the equivariant Tamagawa number conjecture and, as a first application of this approach, we prove new cases of the conjecture over natural families of abelian CM-extensions of totally real fields for which the relevant p-adic L-functions possess trivial zeroes.

Article information

Algebra Number Theory, Volume 11, Number 7 (2017), 1527-1571.

Received: 3 May 2016
Revised: 1 March 2017
Accepted: 10 March 2017
First available in Project Euclid: 12 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11S40: Zeta functions and $L$-functions [See also 11M41, 19F27]
Secondary: 11R23: Iwasawa theory 11R29: Class numbers, class groups, discriminants 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27]

Rubin–Stark conjecture higher-rank Iwasawa main conjecture equivariant Tamagawa number conjecture


Burns, David; Kurihara, Masato; Sano, Takamichi. On Iwasawa theory, zeta elements for $\mathbb{G}_{m}$, and the equivariant Tamagawa number conjecture. Algebra Number Theory 11 (2017), no. 7, 1527--1571. doi:10.2140/ant.2017.11.1527.

Export citation


  • W. Bley, “Wild Euler systems of elliptic units and the equivariant Tamagawa number conjecture”, J. Reine Angew. Math. 577 (2004), 117–146.
  • W. Bley, “Equivariant Tamagawa number conjecture for abelian extensions of a quadratic imaginary field”, Doc. Math. 11 (2006), 73–118.
  • S. Bloch and K. Kato, “$L$-functions and Tamagawa numbers of motives”, pp. 333–400 in The Grothendieck Festschrift, I, edited by P. Cartier et al., Progr. Math. 86, Birkhäuser, Boston, 1990.
  • D. Burns, “Congruences between derivatives of abelian $L$-functions at $s=0$”, Invent. Math. 169:3 (2007), 451–499.
  • D. Burns, “Perfecting the nearly perfect”, Pure Appl. Math. Q. 4:4 (2008), 1041–1058.
  • D. Burns and M. Flach, “On Galois structure invariants associated to Tate motives”, Amer. J. Math. 120:6 (1998), 1343–1397.
  • D. Burns and M. Flach, “Tamagawa numbers for motives with (non-commutative) coefficients”, Doc. Math. 6 (2001), 501–570.
  • D. Burns and C. Greither, “On the equivariant Tamagawa number conjecture for Tate motives”, Invent. Math. 153:2 (2003), 303–359.
  • D. Burns and T. Sano, “On non-abelian zeta elements for $\mathbb{G}_m$”, preprint, 2016,
  • D. Burns, M. Kurihara, and T. Sano, “On zeta elements for $\mathbb{G}_m$”, Doc. Math. 21 (2016), 555–626.
  • D. Burns, M. Kurihara, and T. Sano, “On Stark elements of arbitrary weight and their $p$-adic families”, preprint, 2016.
  • J. Coates and S. Lichtenbaum, “On $l$-adic zeta functions”, Ann. of Math. $(2)$ 98 (1973), 498–550.
  • C. W. Curtis and I. Reiner, Methods of representation theory, vol. I, Wiley, New York, 1981.
  • S. Dasgupta, H. Darmon, and R. Pollack, “Hilbert modular forms and the Gross–Stark conjecture”, Ann. of Math. $(2)$ 174:1 (2011), 439–484.
  • S. Dasgupta, M. Kakde, and K. Ventullo, “On the Gross–Stark conjecture”, preprint, 2016.
  • M. Flach, “The equivariant Tamagawa number conjecture: a survey”, pp. 79–125 in Stark's conjectures: recent work and new directions, edited by D. Burns et al., Contemp. Math. 358, American Mathematical Society, Providence, RI, 2004.
  • M. Flach, “On the cyclotomic main conjecture for the prime $2$”, J. Reine Angew. Math. 661 (2011), 1–36.
  • T. Fukaya and K. Kato, “A formulation of conjectures on $p$-adic zeta functions in noncommutative Iwasawa theory”, pp. 1–85 in Proceedings of the St. Petersburg Mathematical Society, XII, edited by N. N. Uraltseva, Amer. Math. Soc. Transl. Ser. 2 219, American Mathematical Society, Providence, RI, 2006.
  • R. Greenberg, “On a certain $l$-adic representation”, Invent. Math. 21 (1973), 117–124.
  • C. Greither and C. D. Popescu, “An equivariant main conjecture in Iwasawa theory and applications”, J. Algebraic Geom. 24:4 (2015), 629–692.
  • B. H. Gross, “$p$-adic $L$-series at $s=0$”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28:3 (1982), 979–994.
  • K. Kato, “Iwasawa theory and $p$-adic Hodge theory”, Kodai Math. J. 16:1 (1993), 1–31.
  • K. Kato, “Lectures on the approach to Iwasawa theory for Hasse–Weil $L$-functions via $B_{\rm dR}$, I”, pp. 50–163 in Arithmetic algebraic geometry (Trento, 1991), edited by E. Ballico, Lecture Notes in Math. 1553, Springer, Berlin, 1993.
  • M. Kolster, “An idelic approach to the wild kernel”, Invent. Math. 103:1 (1991), 9–24.
  • B. Mazur and K. Rubin, “Refined class number formulas for $\mathbb{G}_m$”, J. Théor. Nombres Bordeaux 28:1 (2016), 185–211.
  • C. D. Popescu, “Integral and $p$-adic refinements of the abelian Stark conjecture”, pp. 45–101 in Arithmetic of $L$-functions, edited by C. D. Popescu et al., IAS/Park City Math. Ser. 18, American Mathematical Society, Providence, RI, 2011.
  • K. Rubin, “The `main conjectures' of Iwasawa theory for imaginary quadratic fields”, Invent. Math. 103:1 (1991), 25–68.
  • K. Rubin, “A Stark conjecture `over $\bf{Z}$' for abelian $L$-functions with multiple zeros”, Ann. Inst. Fourier $($Grenoble$)$ 46:1 (1996), 33–62.
  • T. Sano, “Refined abelian Stark conjectures and the equivariant leading term conjecture of Burns”, Compos. Math. 150:11 (2014), 1809–1835.
  • T. Sano, “On a conjecture for Rubin–Stark elements in a special case”, Tokyo J. Math. 38:2 (2015), 459–476.
  • D. Solomon, “On a construction of $p$-units in abelian fields”, Invent. Math. 109:2 (1992), 329–350.
  • D. Solomon, “Galois relations for cyclotomic numbers and $p$-units”, J. Number Theory 46:2 (1994), 158–178.
  • M. Spiess, “Shintani cocycles and the order of vanishing of $p$-adic Hecke $L$-series at $s=0$”, Math. Ann. 359:1-2 (2014), 239–265.
  • J. Tate, Les conjectures de Stark sur les fonctions $L$ d'Artin en $s=0$, Progress in Mathematics 47, Birkhäuser, Boston, 1984.
  • K. Ventullo, “On the rank one abelian Gross–Stark conjecture”, Comment. Math. Helv. 90:4 (2015), 939–963.
  • A. Wiles, “The Iwasawa conjecture for totally real fields”, Ann. of Math. $(2)$ 131:3 (1990), 493–540.