Invariants and coinvariants of semilocal units modulo elliptic units

Stéphane Viguié

doi:10.5802/jtnb.808

Stéphane Viguié ¹

¹ Université de Franche-Comté 16 route de Gray 25030 Besançon cedex, France

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

Abstract
Résumé

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].

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].

Received: 2011-03-13
Published online: 2012-07-01

MR: 2950704 | Zbl: 1272.11079

DOI: 10.5802/jtnb.808
Author's affiliations:

Stéphane Viguié ¹

¹ Université de Franche-Comté 16 route de Gray 25030 Besançon cedex, France

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

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