Invariants and coinvariants of semilocal units modulo elliptic units

Stéphane Viguié

doi:10.5802/jtnb.808

Stéphane Viguié ¹

¹Université 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 (Original language)
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].

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

Zbl MR

DOI: 10.5802/jtnb.808

Author's affiliations:

Stéphane Viguié ¹

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

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

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