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

Takae Tsuji

doi:10.5802/jtnb.1284

Semi-local units modulo cyclotomic units in the cyclotomic

ℤ_{2}

-extensions

Takae Tsuji ¹

¹ STEM Education Center Tokai University 4-1-1 Kitakaname, Hiratsuka, Kanagawa, Japan

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

Résumé
Abstract

Fixons un corps abélien $k$ dont le conducteur n’est pas divisible par $8$ et notons $k_{\infty} / k$ la $ℤ_{2}$ -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 $ψ$ de $Gal (k / ℚ)$ , nous étudions la structure galoisienne de la $ψ$ -partie $𝒰^{ψ} / 𝒞^{ψ}$ et du $ψ$ -quotient ${(𝒰 / 𝒞)}_{ψ}$ de $𝒰 / 𝒞$ y compris dans le cas $2 ∣ [k : ℚ]$ .

Fix an abelian field $k$ whose conductor is not divisible by $8$ and denote by $k_{\infty} / k$ the cyclotomic $ℤ_{2}$ -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 $ψ$ of $Gal (k / ℚ)$ , we study the Galois module structure of the $ψ$ -part $𝒰^{ψ} / 𝒞^{ψ}$ and $ψ$ -quotient ${(𝒰 / 𝒞)}_{ψ}$ of $𝒰 / 𝒞$ , taking into account the case $2 ∣ [k : ℚ]$ .

Reçu le : 2022-10-27
Révisé le : 2023-01-16
Accepté le : 2023-01-26
Publié le : 2024-11-13

DOI : 10.5802/jtnb.1284

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

Affiliations des auteurs :

Takae Tsuji ¹

¹ STEM Education Center Tokai University 4-1-1 Kitakaname, Hiratsuka, Kanagawa, Japan

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Semi-local units modulo cyclotomic units in the cyclotomic $\mathbb{Z}_2$-extensions},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {445--479},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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     year = {2024},
     doi = {10.5802/jtnb.1284},
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     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1284/}
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Takae Tsuji. Semi-local units modulo cyclotomic units in the cyclotomic $\mathbb{Z}_2$-extensions. Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 2, pp. 445-479. doi : 10.5802/jtnb.1284. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1284/

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