Invariants and coinvariants of semilocal units modulo elliptic units

Viguié, Stéphane

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

Soient $p$ un nombre premier, et $k$ un corps quadratique imaginaire dans lequel $p$ se décompose en deux idéaux maximaux $𝔭$ et $\overline{𝔭}$. Soit $k_{\infty }$ l’unique ${\mathbb{Z}}_p$-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 $\mathrm{L}$ $p$-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 $\overline{𝔭}$. Let $k_{\infty }$ be the unique ${\mathbb{Z}}_p$-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 $\mathrm{L}$-function, as defined in [5].

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