Tohoku Mathematical Journal

A control theorem for the torsion Selmer pointed set

Kenji Sakugawa

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Minhyong Kim defined the Selmer variety associated with a curve $X$ over a number field, which is a non-abelian analogue of the ${\mathbb Q}_p$-Selmer group of the Jacobian variety of $X$. In this paper, we define a torsion analogue of the Selmer variety. Recall that Mazur's control theorem describes the behavior of the torsion Selmer groups of an abelian variety with good ordinary reduction at $p$ in the cyclotomic tower of number fields. We give a non-abelian analogue of Mazur's control theorem by replacing the torsion Selmer group by a torsion analogue of the Selmer variety.

Article information

Tohoku Math. J. (2), Volume 70, Number 2 (2018), 175-223.

First available in Project Euclid: 2 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R23: Iwasawa theory
Secondary: 11R34: Galois cohomology [See also 12Gxx, 19A31]

Selmer variety control theorem Iwasawa theory


Sakugawa, Kenji. A control theorem for the torsion Selmer pointed set. Tohoku Math. J. (2) 70 (2018), no. 2, 175--223. doi:10.2748/tmj/1527904820.

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  • S. Bloch and K. Kato, $L$-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. I, 333–400, Progr. Math., 86, Birkhauser Boston, Boston, MA, 1990.
  • P. Deligne, La conjecture de Weil: II, Publ. Math. IHES 52 (1980), 137–252.
  • M. Demazure and P. Gabriel, Groups Algébriques, Tome I: Géométrie algébrique, généralités, groups commutatifs, Masson & Cie, Editeur, Paris; North-Holland Publishing Co., Amsterdam, 1970.
  • J. M. Fontaine, Le corps des périodes $p$-adiques, with an appendix by Pierre Colmez, Périodes $p$-adiques (Bures-sur-Yvette, 1988), Asterisque No. 223 (1994), 59–111.
  • J. M. Fontaine, Représentations $p$-adiques semistables, Periodes $p$-adiques (Bures-sur-Yvette, 1988), Asterisque No. 223 (1994), 113–184.
  • R. Greenberg, On a certain $l$-adic representation, Invent. Math. 21 (1973), 117–124.
  • B. Ferrero and L. C. Washington, The Iwasawa invariant $\mu_p$ vanishes for abelian number fields, Ann. of Math. (2) 109 (1979), no. 2, 377–395.
  • R. Hain and M. Matsumoto, Weighted completion of Galois groups and Galois actions on the fundamental group of ${\mathbb P}^1-\{0,1,\infty\}$, Compositio Math. 139 (2003), no. 2, 119–167.
  • K. Iwasawa, On $\Gamma$-extensions of algebraic number fields, Bull. Amer. Math. Soc. 65 (1959), 183–226.
  • M. Kim, The motivic fundamental group of ${\mathbb P}^1\setminus\{0,1,\infty\}$ and the theorem of Siegel, Invent. Math. 161 (2005), no. 3, 629–656.
  • M. Kim, The unipotent Albanese map and Selmer varieties for curves, Publ. Res. Inst. Math. Sci. 45 (2009), no. 1, 89–133.
  • M. Kim, Massey products for elliptic curves of rank $1$, J. Amer. Math. Soc. 23 (2010), no. 3, 725–747.
  • N. Katz and W. M. Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73–77.
  • J. P. Labute, On the descending central series of groups with a single defining relation, J. Algebra 14 (1970), 16–23.
  • S. Lang, Complex multiplication, Grundlehren der Mathematischen Wissenschaften 255, Springer-Verlag, New York, 1983.
  • S. Mac Lane, Categories for the working Mathematician, second edition, Graduate Texts in Mathematics, 5 Springer-Verlag, New York, 1998.
  • H. Matsumura, Commutative ring theory, Translated by M. Reid, Second edition, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge, 1989.
  • B. Mazur, Rational Points of Abelian Varieties with Values in Towers of Number Fields, Invent. Math. 18 (1972), 183–266.
  • J. Neukirch, A. Schimidt and K. Wingberg, Cohomology of Number Fields, Grundlehren Math. Wiss. 323, Springer-Verlag, 2000.
  • T. Ochiai, Control Theorem of Bloch–Kato's Selmer Groups for $p$-adic Representations, Jour. of Number theory, 82 (2000), no. 1, 69–90.
  • K. Rubin, Euler systems, Hermann Weyl lectures, Ann. of Math. Studies, vol. 147, Princeton University Press, Princeton, NJ, 2000.
  • R. N. Saavedra, Categories Tannakiennes, Lecture Notes in Mathematics 265, Springer-Verlag, Berlin-New York, 1972.
  • J.-P. Serre, Lie Algebras and Lie Groups, 1964 lectures given at Harvard University, Second edition, Lecture Notes in Mathematics, 1500 Springer-Verlag, Berlin, 1992.
  • J.-P. Serre, Local Fields, Translated from the French by Marvin Jay Greenberg, Graduate Texts in Mathematics, 67. Springer-Verlag, New York-Berlin, 1979.
  • J.-P. Serre, Galois cohomology, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002.
  • J. Tate, Duality theorems in Galois cohomology over number fields, Proc. Intern. Cong. Mathematicians (Stockholm, 1962) 288–295, Inst. Mittag-Leffler, Djursholm, 1963.
  • L. C. Washington, Introduction to cyclotomic fields, Second edition, Graduate Texts in Mathematics, 83 Springer-Verlag, New York, 1997.
  • A. Grothendieck, Séminaire de Géométrie Algébrique du Bois Marie 1960/1961 SGA1, Lecture Notes on Math, 224 Springer Verlag, 1971.