We extend the theory of spinor class fields and relative spinor class fields to study representation problems in several classical linear algebraic groups over number fields. We apply this theory to study the set of isomorphism classes of maximal orders of central simple algebras containing a given maximal Abelian suborder. We also study isometric embeddings of one skew-Hermitian Quaternionic lattice into another.

We find a generator $j_{1,4}$ of the function field on the modular curve X₁(4) by means of classical theta functions θ₂ and θ₃, and estimate the normalized generator $N\left(j_{1,4}\right)$ which becomes the Thompson series of type 4C. With these modular functions we investigate some number theoretic properties.

For any number field $K$ with non-elementary $3$-class group ${\mathrm{Cl}}_3\left(K\right)\simeq C_{3^e}\times C_3$, $e\ge 2$, the punctured capitulation type $\varkappa \left(K\right)$ of $K$ in its unramified cyclic cubic extensions $L_i$, $1\le i\le 4$, is an orbit under the action of $S_3\times S_3$. By means of Artin’s reciprocity law, the arithmetical invariant $\varkappa \left(K\right)$ is translated to the punctured transfer kernel type $\varkappa \left(G_2\right)$ of the automorphism group $G_2=\mathrm{Gal}({\mathrm{F}}_3^2\left(K\right)/K)$ of the second Hilbert $3$-class field of $K$. A classification of finite $3$-groups $G$ with low order and bicyclic commutator quotient $G/G^{\text{'}}\simeq C_{3^e}\times C_3$, $2\le e\le 6$, according to the algebraic invariant...

Let $K=\mathbb{Q}\left(\omega \right)$, with ${\omega }^3=m$ a positive integer, be a pure cubic number field. We show that the elements $\alpha \in K^{\times }$ whose squares have the form $a-\omega$ for rational numbers $a$ form a group isomorphic to the group of rational points on the elliptic curve $E_m:y^2=x^3-m$. This result will allow us to construct unramified quadratic extensions of pure cubic number fields $K$.

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$$...$