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Galois co-descent for étale wild kernels and capitulation

Manfred Kolster, Abbas Movahhedi (2000)

Annales de l'institut Fourier

Let $F$ be a number field with ring of integers $o_{F}$ . For a fixed prime number $p$ and $i \geq 2$ the étale wild kernels $W K_{2 i - 2}^{\overset{´}{e} t} (F)$ are defined as kernels of certain localization maps on the $i$ -fold twist of the $p$ -adic étale cohomology groups of $spec o_{F} [\frac{1}{p}]$ . These groups are finite and coincide for $i = 2$ with the $p$ -part of the classical wild kernel $W K_{2} (F)$ . They play a role similar to the $p$ -part of the $p$ -class group of $F$ . For class groups, Galois co-descent in a cyclic extension $L / F$ is described by the ambiguous class formula given by genus theory....

Galois module structure of rings of integers

Martin J. Taylor (1980)

Annales de l'institut Fourier

Let $E / F$ be a Galois extension of number fields with $Γ =$ Gal $(E / F)$ and with property that the divisors of $(E : F)$ are non-ramified in $E / Q$ . We denote the ring of integers of $E$ by $𝒪_{E}$ and we study $𝒪_{E}$ as a $Z Γ$ -module. In particular we show that the fourth power of the (locally free) class of $𝒪_{E}$ is the trivial class. To obtain this result we use Fröhlich’s description of class groups of modules and his representative for the class of $ℰ_{E}$ , together with new determinantal congruences for cyclic group rings and corresponding congruences...

Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms.

H. Hida (1986)

Inventiones mathematicae

Galois representations of Iwasawa modules

Robert Gold, Manohar Madan (1986)

Acta Arithmetica

Généralisation d’un théorème d’Iwasawa

Jean-François Jaulent (2005)

Journal de Théorie des Nombres de Bordeaux

Nous généralisons à certains quotients finis d’un $Λ$ -module noethérien non nécessairement de torsion le classique théorème d’Iwasawa sur l’expression asymptotique du $ℓ$ -nombre de classes dans les $ℤ_{ℓ}$ -extensions. Puis nous illustrons les résultats obtenus en déterminant explicitement les caractères invariants attachés aux $ℓ$ -groupes de $S$ -classes $T$ -infinitésimales dans une tour cyclotomique à partir de quelques paramètres référents et de données galoisiennes simples des extensions considérées. Un outil...