Invariants and coinvariants of semilocal units modulo elliptic units

Stéphane Viguié

doi:10.5802/jtnb.808

Stéphane Viguié

Journal de Théorie des Nombres de Bordeaux, Tome 24 (2012) no. 2, pp. 487-504.

Résumé
Abstract

Soient $p$ un nombre premier, et $k$ un corps quadratique imaginaire dans lequel $p$ se décompose en deux idéaux maximaux $𝔭$ et $\bar{𝔭}$ . Soit $k_{\infty}$ l’unique $ℤ_{p}$ -extension de $k$ non ramifiée en dehors de $𝔭$ , et soit $K_{\infty}$ une extension finie de $k_{\infty}$ , abélienne sur $k$ . Soit $𝒰_{\infty} / 𝒞_{\infty}$ la limite projective du module des unités semi-locales principales modulo le module des unités elliptiques. Nous prouvons que les différents modules des invariants et des co-invariants de $𝒰_{\infty} / 𝒞_{\infty}$ sont finis. Notre approche utilise les distributions et la fonction $L$ $p$ -adique, définie dans [5].

Let $p$ be a prime number, and let $k$ be an imaginary quadratic number field in which $p$ decomposes into two primes $𝔭$ and $\bar{𝔭}$ . Let $k_{\infty}$ be the unique $ℤ_{p}$ -extension of $k$ which is unramified outside of $𝔭$ , and let $K_{\infty}$ be a finite extension of $k_{\infty}$ , abelian over $k$ . Let $𝒰_{\infty} / 𝒞_{\infty}$ be the projective limit of principal semi-local units modulo elliptic units. We prove that the various modules of invariants and coinvariants of $𝒰_{\infty} / 𝒞_{\infty}$ are finite. Our approach uses distributions and the $p$ -adic $L$ -function, as defined in [5].

Reçu le : 2011-03-13
Publié le : 2012-07-01

MR 2950704 | Zbl 1272.11079

DOI : https://doi.org/10.5802/jtnb.808

@article{JTNB_2012__24_2_487_0,
     author = {St\'ephane Vigui\'e},
     title = {Invariants and coinvariants of semilocal units modulo elliptic units},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {487--504},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {24},
     number = {2},
     year = {2012},
     doi = {10.5802/jtnb.808},
     zbl = {1272.11079},
     mrnumber = {2950704},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.808/}
}

Stéphane Viguié. Invariants and coinvariants of semilocal units modulo elliptic units. Journal de Théorie des Nombres de Bordeaux, Tome 24 (2012) no. 2, pp. 487-504. doi : 10.5802/jtnb.808. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.808/

Bibliographie

[1] J-R. Belliard, Global Units modulo Circular Units: descent without Iwasawa’s Main Conjecture. Canadian J. Math. 61 (2009), 518–533. | MR 2514482

[2] W. Bley, On the Equivariant Tamagawa Number Conjecture for Abelian Extensions of a Quadratic Imaginary Field. Documenta Mathematica 11 (2006), 73–118. | MR 2226270

[3] A. Brumer, On the units of algebraic number fields. Mathematika 14 (1967), 121–124. | MR 220694 | Zbl 0171.01105

[4] J. Coates and A. Wiles, On $p$ -adic $l$ -functions and elliptic units. J. Australian Math. Soc. 26, (1978), 1–25. | MR 510581 | Zbl 0442.12007

[5] E. de Shalit, Iwasawa Theory of Elliptic Curves with Complex Multiplication. Perspectives in Mathematics 3, Academic Press, 1987. | MR 917944 | Zbl 0674.12004

[6] R. Gillard, Fonctions $L$ $p$ -adiques des corps quadratiques imaginaires et de leurs extensions abéliennes. J. Reine Angew. Math. 358 (1985), 76–91. | MR 797675 | Zbl 0551.12011

[7] C. Greither, Class groups of abelian fields, and the main conjecture. Annal. Inst. Fourier 42 (1992), 445–499. | Numdam | MR 1182638 | Zbl 0729.11053

[8] K. Iwasawa, Lectures on $p$ -adic $L$ -functions. Princeton University Press, 1972. | MR 360526 | Zbl 0236.12001

[9] S. Lang, Cyclotomic Fields I and II. Springer-Verlag, 1990. | MR 1029028 | Zbl 0395.12005

[10] Neukirch, Schmidt and Wingberg, Cohomology of Number Fields. Springer-Verlag, 2000. | MR 1737196

[11] H. Oukhaba, On Iwasawa theory of elliptic units and $2$ -ideal class groups. Prépub. lab. Math. Besançon (2010).

[12] G. Robert, Unités elliptiques. Bull. soc. math. France 36, (1973). | Numdam | MR 469889 | Zbl 0314.12006

[13] G. Robert, Unités de Stark comme unités elliptiques. Prépub. Inst. Fourier 143, (1989).

[14] G. Robert, Concernant la relation de distribution satisfaite par la fonction $ϕ$ associée à un réseau complexe. Inven. math. 100 (1990), 231–257. | MR 1047134 | Zbl 0729.11029

[15] K. Rubin, The ”main conjectures” of Iwasawa theory for imaginary quadratic fields. Inven. math. 103 (1991), 25–68. | MR 1079839 | Zbl 0737.11030

[16] K. Rubin, More ”Main Conjectures” for Imaginary Quadratic Fields. Centre Rech. Math. 4 (1994), 23–28. | MR 1260952 | Zbl 0809.11066

[17] S. Viguié, On the classical main conjecture for imaginary quadratic fields. Prépub. lab. Math. Besançon (2011).

[18] S. Viguié, Global units modulo elliptic units and ideal class groups. to appear in Int. J. Number Theory.