Tokyo Journal of Mathematics

On Zeta Elements and Refined Abelian Stark Conjectures


Full-text: Access denied (no subscription detected)

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 apply recent methods of Burns, Kurihara and Sano in [5] to study connections between the values at \(s=0\) of the higher derivatives of abelian \(L\)-functions of number fields and the higher Fitting ideals of the canonical Selmer groups of \(\mathbb{G}_m\). Whereas Burns, Kurihara and Sano apply these methods to the setting of the `Rubin-Stark conjecture', we study the `evaluators' defined in a more general setting by Emmons and Popescu in [7] and by Vallieres in [14]. This allows us to conjecture that the ideals formed from the images of the evaluators can be described precisely in terms of the higher Fitting ideals of the canonical Selmer groups of \(\mathbb{G}_m\). Moreover, we are able to prove that this conjecture follows from the equivariant Tamagawa number conjecture.

Article information

Tokyo J. Math., Volume 40, Number 1 (2017), 123-151.

First available in Project Euclid: 8 August 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


LIVINGSTONE BOOMLA, Alice. On Zeta Elements and Refined Abelian Stark Conjectures. Tokyo J. Math. 40 (2017), no. 1, 123--151. doi:10.3836/tjm/1502179219.

Export citation


  • D. Burns, Congruences between derivatives of abelian (L)-functions at (s=0), Invent. Math. 169 (2007), 451–499.
  • D. Burns, On derivatives of (p)-adic (L)-series at (s=0), Available from:
  • D. Burns and M. Flach, Tamagawa numbers for motives with (non-commutative) coefficients, Documenta Math. 6 (2001), 501–570.
  • D. Burns and C. Greither, On the equivariant Tamagawa number conjecture for Tate motives, Invent. Math. 154 (2003), 303–359.
  • D. Burns, M. Kurihara and T. Sano, On zeta elements for (\mathbbG_m), Documenta Math. 21 (2016), 555–626.
  • C. W. Curtis and I. Reiner, Methods of Representation Theory, Vol. I, John Wiley and Sons, New York, 1987.
  • C. Emmons and C. Popescu, Special values of abelian (L)-functions at (s = 0), J. Number Theory 129 (2009), 1350–1365.
  • M. Flach, On the cyclotomic main conjecture for the prime 2, J. reine angew. Math. 661 (2011), 1–36.
  • D.G. Northcott, Finite Free Resolutions, Cambridge Univ. Press, Cambridge, New York, 1976.
  • H. Stark, (L)-functions at s=1 I, II, III, IV, Advances in Math. 7 (1971), 301–343, 17 (1975), 60–92, 22 (1976), 64–84, 35 (1980), 197–235.
  • J. Tate, On Stark's conjectures on the behavior of (L(s, \chi)) at (s=0), J. Fac. Sci. Univ. Tokyo 28 (1982), 963–978.
  • J. Tate, Les Conjectures de Stark sur les Fonctions (L) d'Artin en (s=0) (notes par D. Bernardi et N. Schappacher), Progress in Math., 47, Birkhäuser, Boston, 1984.
  • D. Vallières, The equivariant Tamagawa number conjecture and the extended abelian Stark conjecture, Preprint, Accepted for publication in J. reine angew. Math.