Euler Systems

  • V. A. Kolyvagin
Part of the Modern Birkhäuser Classics book series (MBC)


In this paper we study Euler systems defined by the characterizing condition AX1, perhaps with the addition of other conditions (AX2 and AX3 systems, see §1). Our main purpose is to apply them to determine the structure of the class groups of certain algebraic number fields R, and the Mordell-Weil groups and Shafarevich-Tate groups of Weil curves. In the case of the class group Cl of a field R, Theorem 7 of §2 says that, if the Galois group G of R is annihilated by l − 1, where l is a rational prime, and if ψ is a homomorphism from G to the group of (l — l)-th roots of unity in Z l, then (under certain conditions on R and ψ) any Euler system associated to R which is non-degenerate (in its (l, ψ)-component) determines the structure of the ψ-component of ClZ l, i.e., it determines the set of integers n i, n in i+1, such that \( (Cl \otimes Z_l )_\psi \simeq \sum\nolimits_{i = 1}^{i_0 } {Z/l^{n,} } \) as an abelian group. Theorem 7 also shows how the Euler system determines bases of (ClZ l)ψ consisting of prime divisor classes, the expansions of certain prime divisor classes in these bases, and also certain representations of primary numbers. For example, this holds for the cyclotomic field K l = Ql) (see below) with odd characters ψ and the system of Gauss sums, or with even characters ψ and the system of cyclotomic units. As a corollary we find that the order of X = (ClZ l)ψ is bounded from above by the predicted explicit order [X]?; and this, along with formulas for the class number, enables us in several cases (cyclotomic fields, fields which are abelian extensions of an imaginary quadratic field) to prove that [X] and [X]? are equal.


Elliptic Curve Prime Divisor Multiplicative Group Infinite Order Euler System 
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  1. [1]
    Z. I. Borevich, I. R. Shafarevich, Number Theory, Academic Press, 1966.Google Scholar
  2. [2]
    K. Ribet, “A modular construction of unramified p-extensions of Q(μ p)”, Invent. Math., 34 (1976), 151–162.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    A. Wiles, “Modular curves and the class group of QP)”, Invent. Math., 58 (1980), 1–35.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    B. Mazur, A. Wiles, “Class fields of abelian extensions of Q”, Invent. Math., 76 (1984), 179–330.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    A. Wiles, “On p-adic representations for totally real fields”, Ann. Math., 123 (1986), No. 3, 407–456.CrossRefMathSciNetGoogle Scholar
  6. [6]
    F. Thaine, “On the ideal class groups of real abelian number fields”, to appear in Ann. of Math.Google Scholar
  7. [7]
    K. Rubin, “Global units and ideal class groups”, Invent. Math., 89 (1987), 511–526.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    K. Rubin, “Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication”, Invent. Math., 89 (1987), 527–560.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    J. Coates, A. Wiles, “On the conjecture of Birch and Swinnerton-Dyer”, Invent. Math., 39 (1977), 223–251.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    B. H. Gross, D. B. Zagier, “Heegner points and derivatives of L-series”, Invent. Math., 84 (1986), 225–320.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    V. A. Kolyvagin, “Finiteness of E(Q) and III(E, Q) for a subclass of Weil curves”, Izvestiya AN SSSR, Ser. Mat., 52 (1988), No. 3, 522–540.MathSciNetGoogle Scholar
  12. [12]
    V. A. Kolyvagin, “On the Mordell-Weil group and the Shafarevich-Tate group of Weil elliptic curves”, Izvestiya AN SSSR, Ser. Mat., 52 (1988), No. 6.Google Scholar
  13. [13]
    D. B. Zagier, “Modular points, modular curves, modular surfaces and modular forms”, in Arbeitstagung Bonn 1984, Springer Lecture Notes in Math., 1111 (1985), 225–248.CrossRefMathSciNetGoogle Scholar
  14. [14]
    G. Stevens, Arithmetic on Modular Curves, Birkhäuser, 1982.Google Scholar
  15. [15]
    B. Mazur, H. Swinnerton-Dyer, “Arithmetic of Weil curves”, Invent. Math., 25 (1974), 1–61.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    B. Perrin-Riou, “Points de Heegner et derivées de fonctions L p-adiques”, Invent. Math., 89 (1987), 455–510.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    S. Lang, Elliptic Functions, Addison-Wesley, 1973.Google Scholar
  18. [18]
    K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, 1982.Google Scholar
  19. [19]
    M. I. Bashmakov, “The cohomology of abelian varieties over a number field”, Uspekhi Mat. Nauk, 27 (1972), No. 6, 25–66 [English translation: Russian Math. Surveys, 27 (1972), No. 6, 25–70].Google Scholar

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© Birkhäuser Boston 2007

Authors and Affiliations

  • V. A. Kolyvagin
    • 1
  1. 1.Steklov InstituteMoscowU.S.S.R.

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