Let $K$ be a quadratic imaginary number field of discriminant $d$. For $t\in \mathbb{N}$ let ${𝔒}_t$ denote the order of conductor $t$ in $K$ and $j\left({𝔒}_t\right)$ its modular invariant which is known to generate the ring class field modulo $t$ over $K$. The coefficients of the minimal equation of $j\left({𝔒}_t\right)$ being quite large Weber considered in [We] the functions $f,f_1,f_2,{\gamma }_2,{\gamma }_3$ defined below and thereby obtained simpler generators of the ring class fields. Later on the singular values of these functions played a crucial role in Heegner’s solution [He] of the class...

Let $h_n$ denote the class number of $n$-th layer of the cyclotomic ${\mathbb{Z}}_2$-extension of $\mathbb{Q}$. Weber proved that $h_n(n\ge 1)$ is odd and Horie proved that $h_n(n\ge 1)$ is not divisible by a prime number $\ell$ satisfying $\ell \equiv 3,5(mod8)$. In a previous paper, the authors showed that $h_n(n\ge 1)$ is not divisible by a prime number $\ell$ less than ${10}^7$. In this paper, by investigating properties of a special unit more precisely, we show that $h_n(n\ge 1)$ is not divisible by a prime number $\ell$ less than $1.2·{10}^8$. Our argument also leads to the conclusion that $h_n(n\ge 1)$ is not divisible by a prime number...

Let $\epsilon$ be an algebraic unit of the degree $n\ge 3$. Assume that the extension $\mathbb{Q}\left(\epsilon \right)/\mathbb{Q}$ is Galois. We would like to determine when the order $\mathbb{Z}\left[\epsilon \right]$ of $\mathbb{Q}\left(\epsilon \right)$ is $\mathrm{Gal}\left(\mathbb{Q}\right(\epsilon )/\mathbb{Q})$-invariant, i.e. when the $n$ complex conjugates ${\epsilon }_1,\cdots ,{\epsilon }_n$ of $\epsilon$ are in $\mathbb{Z}\left[\epsilon \right]$, which amounts to asking that $\mathbb{Z}[{\epsilon }_1,\cdots ,{\epsilon }_n]=\mathbb{Z}\left[\epsilon \right]$, i.e., that these two orders of $\mathbb{Q}\left(\epsilon \right)$ have the same discriminant. This problem has been solved only for $n=3$ by using an explicit formula for the discriminant of the order $\mathbb{Z}[{\epsilon }_1,{\epsilon }_2,{\epsilon }_3]$. However, there is no known similar formula for $n>3$. In the present paper, we put forward and motivate three...

Sia a un intero algebrico con il polinomio minimale $f\left(X\right)$. Si danno condizioni necessarie e sufficienti affinché l'anello $\mathbb{Z}\left[\alpha \right]$ sia seminormale o $t$-chiuso per mezzo di $f\left(X\right)$. Come applicazione, in particolare, si ottiene che se $f\left(X\right)=X^3+aX+b$, $a$, $b\in \mathbb{Z}$ le condizioni sono espresse mediante il discriminante de $f\left(X\right)$.

The intrinsic algebraic entropy ent(ɸ) of an endomorphism ɸ of an Abelian group G can be computed using fully inert subgroups of ɸ-invariant sections of G, instead of the whole family of ɸ-inert subgroups. For a class of groups containing the groups of finite rank, aswell as those groupswhich are trajectories of finitely generated subgroups, it is proved that only fully inert subgroups of the group itself are needed to comput ent(ɸ). Examples show how the situation may be quite different outside...

The theory of Whittaker functors for $G{L}_n$ is an essential technical tools in Gaitsgory’s proof of the Vanishing Conjecture appearing in the geometric Langlands correspondence. We define Whittaker functors for $G𝕊p_4$ and study their properties. These functors correspond to the maximal parabolic subgroup of $G𝕊p_4$, whose unipotent radical is not commutative.We also study similar functors corresponding to the Siegel parabolic subgroup of $G𝕊p_4$, they are related with Bessel models for $G𝕊p_4$ and Waldspurger models for $G{L}_2$.We...

In this article we will describe a surprising observation that occurred in the construction of quadratic unramified extensions of a family of pure cubic number fields. Attempting to find an explanation will lead us on a magical mystery tour through the land of pure cubic number fields, Hilbert class fields, and elliptic curves.

Let K be a number field. Assume that the 2-rank of the ideal class group of K is equal to the 2-rank of the narrow ideal class group of K. Moreover, assume K has a unique dyadic prime and the class of is a square in the ideal class group of K. We prove that if ₁,...,ₙ are finite primes of K such that ∙ the class of ${}_i$ is a square in the ideal class group of K for every i ∈ 1,...,n, ∙ -1 is a local square at ${}_i$ for every nondyadic ${}_i\in ₁,...,ₙ$, then ₁,...,ₙ is the wild set of some self-equivalence of the field...