Semi-local units modulo cyclotomic units in the cyclotomic ${\mathbb{Z}}_2$-extensions

Tsuji, Takae

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 2, pp. 445-479

Fix an abelian field $k$ whose conductor is not divisible by $8$ and denote by $k_{\infty }/k$ the cyclotomic ${\mathbb{Z}}_2$-extension with $n$-th layer $k_n$. Let $𝒰$ (resp. $𝒞$) be the projective limit of the semi-local units at $2$ (resp. of the cyclotomic units) of $k_n$. For a non-trivial even character $\psi$ of $\mathrm{Gal}(k/\mathbb{Q})$, we study the Galois module structure of the $\psi$-part ${𝒰}^{\psi }/{𝒞}^{\psi }$ and $\psi$-quotient ${(𝒰/𝒞)}_{\psi }$ of $𝒰/𝒞$, taking into account the case $2\mid [k:\mathbb{Q}]$.

Fixons un corps abélien $k$ dont le conducteur n’est pas divisible par $8$ et notons $k_{\infty }/k$ la ${\mathbb{Z}}_2$-extension cyclotomique avec le $n$-ième corps intermédiaire $k_n$. Soit $𝒰$ (resp. $𝒞$) la limite projective des groupes des unités semi-locales (resp. des unités cyclotomiques) en 2 de $k_n$. Pour un caractère pair non-trivial $\psi$ de $\mathrm{Gal}(k/\mathbb{Q})$, nous étudions la structure galoisienne de la $\psi$-partie ${𝒰}^{\psi }/{𝒞}^{\psi }$ et du $\psi$-quotient ${(𝒰/𝒞)}_{\psi }$ de $𝒰/𝒞$ y compris dans le cas $2\mid [k:\mathbb{Q}]$.

Classification : 11R23
Keywords: Iwasawa theory, cyclotomic units, $p$-adic $L$-functions

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