Proceedings of the Japan Academy, Series A, Mathematical Sciences

A note on the Bloch-Tamagawa space and Selmer groups

Niranjan Ramachandran

Full-text: Open access


For any abelian variety $A$ over a number field, we construct an extension of the Tate-Shafarevich group by the Bloch-Tamagawa space using the recent work of Lichtenbaum and Flach. This gives a new example of a Zagier sequence for the Selmer group of $A$.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 91, Number 5 (2015), 61-65.

First available in Project Euclid: 30 April 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10] 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx] 20G30: Linear algebraic groups over global fields and their integers 22E41: Continuous cohomology [See also 57R32, 57Txx, 58H10]

Abelian varieties L-functions Tamagawa numbers Selmer groups


Ramachandran, Niranjan. A note on the Bloch-Tamagawa space and Selmer groups. Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 5, 61--65. doi:10.3792/pjaa.91.61.

Export citation


  • A. Asok, B. Doran and F. Kirwan, Yang-Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics, Bull. Lond. Math. Soc. 40 (2008), no. 4, 533–567.
  • S. Bloch, A note on height pairings, Tamagawa numbers, and the Birch and Swinnerton-Dyer conjecture, Invent. Math. 58 (1980), no. 1, 65–76.
  • S. Bloch and K. Kato, $L$-functions and Tamagawa numbers of motives, in The Grothendieck Festschrift, Vol. I, Progr. Math., 86, Birkhäuser, Boston, MA, 1990, pp. 333–400.
  • P. Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77.
  • M. Flach, Cohomology of topological groups with applications to the Weil group, Compos. Math. 144 (2008), no. 3, 633–656.
  • K. Iwasawa, On the rings of valuation vectors, Ann. of Math. (2) 57 (1953), 331–356.
  • K. Iwasawa, Letter to J. Dieudonné, in Zeta functions in geometry (Tokyo, 1990), 445–450, Adv. Stud. Pure Math., 21, Kinokuniya, Tokyo, 1992.
  • R. E. Kottwitz, Tamagawa numbers, Ann. of Math. (2) 127 (1988), no. 3, 629–646.
  • S. Lichtenbaum, Euler characteristics and special values of zeta-functions, in Motives and algebraic cycles, Fields Inst. Commun., 56, Amer. Math. Soc., Providence, RI, 2009, pp. 249–255.
  • S. Lichtenbaum, The Weil-étale topology for number rings, Ann. of Math. (2) 170 (2009), no. 2, 657–683.
  • J. Oesterlé, Construction de hauteurs archimédiennes et $p$-adiques suivant la methode de Bloch, in Seminar on Number Theory, Paris 1980–81 (Paris, 1980/1981), 175–192, Progr. Math., 22, Birkhäuser, Boston, MA, 1982.
  • T. Ono, On Tamagawa numbers, in Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), 122–132, Amer. Math. Soc., Providence, RI, 1996.
  • A. J. Scholl, Extensions of motives, higher Chow groups and special values of $L$-functions, in Séminaire de Théorie des Nombres, Paris, 1991–92, Progr. Math., 116, Birkhäuser, Boston, MA, 1993, pp. 279–292.
  • A. J. Scholl, Height pairings and special values of $L$-functions, in Motives (Seattle, WA, 1991), 571–598, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.
  • T. Tamagawa, Adèles, in Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), 113–121, Amer. Math. Soc., Providence, RI, 1966.
  • J. T. Tate, Fourier analysis in number fields, and Hecke's zeta-functions, in Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), 305–347, Thompson, Washington, DC, 1967.
  • B. Wieland (, Why are tamagawa numbers equal to Pic/Sha? MathOverflow. URL: (version: 2010-10-31).
  • D. Zagier, The Birch-Swinnerton-Dyer conjecture from a naive point of view, in Arithmetic algebraic geometry (Texel, 1989), 377–389, Progr. Math., 89, Birkhäuser, Boston, MA, 1991.