In this paper, first we introduce a new notion of commuting condition that $\phi {\phi }_{1}A=A{\phi }_1\phi$ between the shape operator $A$ and the structure tensors $\phi$ and ${\phi }_1$ for real hypersurfaces in $G_2\left({ℂ}^{m+2}\right)$. Suprisingly, real hypersurfaces of type $\left(A\right)$, that is, a tube over a totally geodesic $G_2\left({ℂ}^{m+1}\right)$ in complex two plane Grassmannians $G_2\left({ℂ}^{m+2}\right)$ satisfy this commuting condition. Next we consider a complete classification of Hopf hypersurfaces in $G_2\left({ℂ}^{m+2}\right)$ satisfying the commuting condition. Finally we get a characterization of Type $\left(A\right)$ in terms of such commuting...

We begin by giving a criterion for a number field $K$ with 2-class group of rank 2 to have a metacyclic Hilbert 2-class field tower, and then we will determine all real quadratic number fields $\mathbb{Q}\left(\sqrt{d}\right)$ that have a metacyclic nonabelian Hilbert $2$-class field tower.

In questo lavoro vengono studiati gli zeri reali di una classe di serie di Dirichlet, che generalizzano le funzioni $L(s,\chi )$, definite in [8], Combinando le tecniche elementari di Pintz [9] con alcuni metodi analitici si ottiene l’estensione dei classici teoremi di Hecke e Siegel.

This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of real zeros as the weight goes to infinity.

We study so-called real zeros of holomorphic Hecke cusp forms, that is, zeros on three geodesic segments on which the cusp form (or a multiple of it) takes real values. Ghosh and Sarnak, who were the first to study this problem, showed the existence of many such zeros if many short intervals contain numbers whose prime factors all belong to a certain subset of the primes.We prove new results concerning this sieving problem which leads to improved lower bounds for the number of real zeros.

Dans cet article, on montre de manière explicite que toute forme $\mathbb{Z}$-bilinéaire symétrique non dégénérée de rang pair, et non $\mathbb{Q}$-isomorphe au plan hyperbolique, se réalise comme forme trace hermitienne amplifiée d’une algèbre $\mathbb{Z}\left[\alpha \right]$, où $\alpha$ est un entier algébrique. Plus précisemment, on montre que pour tout $S\in M_{2n}\left(\mathbb{Z}\right)$ symétrique, avec det$S\ne 0$ (et det$S\neg \equiv -1$ (mod ${\mathbb{Q}}^{*2}$) si $n=1$), il existe un entier algébrique $\alpha$, une involution $\mathbb{Q}$-linéaire $\sigma$ de $\mathbb{Q}\left(\alpha \right),\lambda \in \mathbb{Q}\left(\alpha \right)\sigma$-symétrique et une $\mathbb{Z}$-base $v_1,\cdots ,v_{2n}$ d’un idéal de $\mathbb{Z}\left[\alpha \right]$ tels que $S=\left(T{r}_{\mathbb{Q}\left(\alpha \right)/\mathbb{Q}}\left(\lambda v_i{v}_j^{\sigma }\right)\right)$.

Given an algebraic number field $k$ and a finite group $\Gamma$, we write $ℛ\left(O_k\left[\Gamma \right]\right)$ for the subset of the locally free classgroup $\mathrm{Cl}\left(O_k\left[\Gamma \right]\right)$ consisting of the classes of rings of integers $O_N$ in tame Galois extensions $N/k$ with $\mathrm{Gal}(N/k)\cong \Gamma$. We determine $ℛ\left(O_k\left[\Gamma \right]\right)$, and show it is a subgroup of $\mathrm{Cl}\left(O_k\left[\Gamma \right]\right)$ by means of a description using a Stickelberger ideal and properties of some cyclic codes, when $k$ contains a root of unity of prime order $p$ and $\Gamma =V⋊C$, where $V$ is an elementary abelian group of order $p^r$ and $C$ is a cyclic group of order $m\>1$ acting faithfully on...