Tohoku Mathematical Journal

On the two-variables main conjecture for extensions of imaginary quadratic fields

Stéphane Vigué

Full-text: Open access

Abstract

Let $p$ be a prime number at least 5, and let $k$ be an imaginary quadratic number field in which $p$ decomposes into two conjugate primes. Let $k_\infty$ be the unique ${\boldsymbol Z}_p^2$-extension of $k$, and let $K_\infty$ be a finite extension of $k_\infty$, abelian over $k$. We prove that in $K_\infty$, the characteristic ideal of the projective limit of the $p$-class group coincides with the characteristic ideal of the projective limit of units modulo elliptic units. Our approach is based on Euler systems, which were first used in this context by Rubin.

Article information

Source
Tohoku Math. J. (2), Volume 65, Number 3 (2013), 441-465.

Dates
First available in Project Euclid: 12 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1378991025

Digital Object Identifier
doi:10.2748/tmj/1378991025

Mathematical Reviews number (MathSciNet)
MR3102544

Subjects
Primary: 11G16: Elliptic and modular units [See also 11R27]
Secondary: 11R23: Iwasawa theory 11R65: Class groups and Picard groups of orders

Keywords
Elliptic units Euler systems Iwasawa theory

Citation

Vigué, Stéphane. On the two-variables main conjecture for extensions of imaginary quadratic fields. Tohoku Math. J. (2) 65 (2013), no. 3, 441--465. doi:10.2748/tmj/1378991025. https://projecteuclid.org/euclid.tmj/1378991025


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References

  • W. Bley, Equivariant Tamagawa number conjecture for abelian extensions of a quadratic imaginary field, Doc. Math. 11 (2006), 73–118.
  • E. de Shalit, Iwasawa theory of elliptic curves with complex multiplication. $p$-adic $L$ functions, Perspect. Math. 3, Academic Press, Inc., Boston, MA, 1987.
  • R. Gillard, Fonctions $L$ $p$-adiques des corps quadratiques imaginaires et de leurs extensions abéliennes, J. Reine Angew. Math. 358 (1985), 76–91.
  • R. Greenberg, On the structure of certain Galois groups, Invent. Math. 47 (1978), 85–99.
  • C. Greither, Class groups of abelian fields, and the main conjecture, Ann. Inst. Fourier (Grenoble) 42 (1992), 445–499.
  • K. Iwasawa, On ${\boldsymbol Z}_l$-extensions of algebraic number fields, Ann. of Math. 98 (1973), 246–326.
  • A. Jilali and H. Oukhaba, Stark units in ${\boldsymbol Z}_p$-extensions, Funct. Approx. Comment. Math. 45 (2011), part I, 105–124.
  • J. Johnson-Leung and G. Kings, On the equivariant main conjecture for imaginary quadratic fields, J. Reine Angew. Math. 653 (2011), 75–114.
  • H. Oukhaba, On Iwasawa theory of elliptic units and 2-ideal class groups, J. Ramanujan Math. Soc. 27 (2012), no. 3, 255–373.
  • H. Oukhaba and S. Viguié, The Gras conjecture in function fields by Euler systems, Bull. Lond. Math. Soc. 43 (2011), 523–535.
  • B. Perrin-Riou, Arithmétique des courbes elliptiques et théorie d'Iwasawa, Supplément au Bulletin de la société mathématique de France 112 n$^{\circ}4$ Mém. Soc. Math. France (N.S.) No. 17 (1984), 130 pp.
  • G. Robert, Unités elliptiques, Bull. Soc. Math. France No 36, Bull. Soc. Math. France, Tome 101, Société Mathématique de France, Paris, 1973.
  • G. Robert, Unités de Stark comme unités elliptiques, Prépublication de l'Institut Fourier 143, 1989.
  • G. Robert, Concernant la relation de distribution satisfaite par la fonction $\varphi$ associée à un réseau complexe, Invent. Math. 100 (1990), 231–257.
  • K. Rubin, On the main conjecture of Iwasawa theory for imaginary quadratic fields, Invent. Math. 93 (1988), 701–713.
  • K. Rubin, The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), 25–68.
  • K. Rubin, More “main conjectures” for imaginary quadratic fields. Elliptic curves and related topics, 23–28, CRM Proc. Lecture Notes 4, Amer. Math. Soc., Providence, RI, 1994.
  • J. Tate, Les conjectures de Stark sur les fonctions $\mathrm{L}$ d'Artin en $s=0$, Lecture notes edited by Dominique Bernardi and Norbert Schappacher, Progr. Math. 47, Birkhäuser Boston, Inc., Boston, MA, 1984.
  • S. Viguié, On the classical main conjecture for imaginary quadratic fields, to appear in JP J. Algebra Number Theory Appl.
  • S. Viguié, Global units modulo elliptic units and ideal class groups, Int. J. Number Theory 8 (2012), 569–588.
  • J.-P. Wintenberger, Structure galoisienne de limites projectives d'unités locales, Compositio Math. 42 (1980/81), 89–103.