If $K/k$ is a finite Galois extension of number fields with Galois group $G$, then the kernel of the capitulation map $C{l}_k\to C{l}_K$ of ideal class groups is isomorphic to the kernel $X\left(H\right)$ of the transfer map $H/H^{\text{'}}\to A,$ where $H=\text{Gal}(\tilde{K}/k),A=\text{Gal}(\tilde{K}/K)$ and $\tilde{K}$ is the Hilbert class field of $K$. H. Suzuki proved that when $G$ is abelian, $\left|G\right|$ divides $\left|X\right(H\left)\right|$. We call a finite abelian group $X$ a transfer kernel for $G$ if $X\cong X\left(H\right)$ for some group extension $A\hookrightarrow H\twoheadrightarrow G$. After characterizing transfer kernels in terms of integral representations of $G$, we show that $X$ is a transfer kernel for...

Let $K=k\left(\sqrt{-p\epsilon \sqrt{l}}\right)$ with $k=\mathbb{Q}\left(\sqrt{l}\right)$ where $l$ is a prime number such that $l=2$ or $l\equiv 5mod8$, $\epsilon$ the fundamental unit of $k$, $p$ a prime number such that $p\equiv 1mod4$ and ${\left(\frac{p}{l}\right)}_4=-1$, $K_2^{\left(1\right)}$ the Hilbert $2$-class field of $K$, $K_2^{\left(2\right)}$ the Hilbert $2$-class field of $K_2^{\left(1\right)}$ and $G=Gal(K_2^{\left(2\right)}/K)$ the Galois group of $K_2^{\left(2\right)}/K$. According to E. Brown and C. J. Parry [7] and [8], $C_{2,K}$, the Sylow $2$-subgroup of the ideal class group of $K$, is isomorphic to $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, consequently $K_2^{\left(1\right)}/K$ contains three extensions $F_i/K$$...$

Soient $\mathbf{K}=\mathbf{Q}\left(\sqrt{-pq(2+\sqrt{2})}\right)$ où $p$ et $q$ deux nombres premiers différents tels que $p\equiv q\equiv \pm 5mod8$, ${\mathbf{K}}_2^{\left(1\right)}$ le $2$-corps de classes de Hilbert de $\mathbf{K}$, ${\mathbf{K}}_2^{\left(2\right)}$ le $2$-corps de classes de Hilbert de ${\mathbf{K}}_2^{\left(1\right)}$ et $G$ le groupe de Galois de ${\mathbf{K}}_2^{\left(2\right)}/K$. D’après [4], la $2$-partie $C_{2,\mathbf{K}}$ du groupe de classes de $\mathbf{K}$ est de type $(2,2)$, par suite ${\mathbf{K}}_2^{\left(1\right)}$ contient trois extensions ${\mathbf{F}}_i/\mathbf{K}$ ; $i=1,2,3$. Dans ce papier, on s’interesse au problème de capitulation des $2$-classes d’idéaux de $\mathbf{K}$ dans ${\mathbf{F}}_i$$...$

In this article we obtain class invariants and cyclotomic unit groups by considering specializations of modular units. We construct these modular units from functional solutions to higher order $q$-recurrence equations given by Selberg in his work generalizing the Rogers-Ramanujan identities. As a corollary, we provide a new proof of a result of Zagier and Gupta, originally considered by Gauss, regarding the Gauss periods. These results comprise part of the author’s 2006 Ph.D. thesis [6] in which...

We apply the Shimura reciprocity law to determine when values of modular functions of higher level can be used to generate the Hilbert class field of an imaginary quadratic field. In addition, we show how to find the corresponding polynomial in these cases. This yields a proof for conjectural formulas of Morain and Zagier concerning such polynomials.

Pour $l$ premier impair, l’étude du $l$-groupe des classes d’idéaux des extensions abéliennes de degré premier à $l$ se ramène à celle de groupes notés ${\mathbf{H}}_{\varphi }$, où $\varphi$ parcourt un certain ensemble de caractères $l$-adiques irréductibles.Il est démontré, dans cet article, une généralisation des congruences de Leopoldt et Fresnel entre les fonctions $L_{l}l$-adiques et les nombres de Bernoulli généralisés. Cette généralisation conduit à une amélioration de la connaissance des ${\mathbf{H}}_{\varphi }$ : en effet, la juxtaposition de ce résultat...

Using both class field and Kummer theories, we propose calculations of orders of two Selmer groups, and compare them: the quotient of the orders only depends on local criteria.