Let $(𝒪,\Sigma ,F_{\infty })$ be an arithmetic ring of Krull dimension at most 1, $𝒮=\mathrm{Spec}𝒪$ and $(\pi :𝒳\to 𝒮;{\sigma }_1,...,{\sigma }_n)$ an $n$-pointed stable curve of genus $g$. Write $𝒰=𝒳\setminus {\cup }_j{\sigma }_j\left(𝒮\right)$. The invertible sheaf ${\omega }_{𝒳/𝒮}({\sigma }_1+\cdots +{\sigma }_n)$ inherits a hermitian structure ${\parallel ·\parallel }_{\mathrm{hyp}}$ from the dual of the hyperbolic metric on the Riemann surface ${𝒰}_{\infty }$. In this article we prove an arithmetic Riemann-Roch type theorem that computes the arithmetic self-intersection of ${\omega }_{𝒳/𝒮}{({\sigma }_1+...+{\sigma }_n)}_{\mathrm{hyp}}$. The theorem is applied to modular curves $X\left(\Gamma \right)$, $\Gamma ={\Gamma }_0\left(p\right)$ or ${\Gamma }_1\left(p\right)$, $p\ge 11$ prime, with sections given by the cusps. We show $Z^{\text{'}}(Y\left(\Gamma \right),1)\sim e^a{\pi }^b{\Gamma }_2{(1/2)}^{c}L(0,{ℳ}_{\Gamma })$, with $p\equiv 11mod12$ when $\Gamma ={\Gamma }_0\left(p\right)$. Here $Z\left(Y\right(\Gamma ),s)$ is the Selberg...

Let $K/\mathbb{Q}$ be a nonnormal cubic extension which is given by an irreducible polynomial $g\left(x\right)=x^3+a{x}^2+bx+c$. Denote by ${\zeta }_K\left(s\right)$ the Dedekind zeta-function of the field $K$ and $a_K\left(n\right)$ the number of integral ideals in $K$ with norm $n$. In this note, by the higher integral mean values and subconvexity bound of automorphic $L$-functions, the second and third moment of $a_K\left(n\right)$ is considered, i.e., $$\sum _{n\le x}{a}_K^2\left(n\right)=x{P}_1(logx)+O\left(x^{5/7+ϵ}\right),\sum _{n\le x}{a}_K^3\left(n\right)=x{P}_4(logx)+O\left(X^{321/356+ϵ}\right),$$ where $P_1\left(t\right)$, $P_4\left(t\right)$ are polynomials of degree 1, 4, respectively, $ϵ>0$ is an arbitrarily small number.

Let $k$ be a $p$-adic field. Let $G$ be the group of $k$-rational points of a connected reductive group $𝖦$ defined over $k$, and let $𝔤$ be its Lie algebra. Under certain hypotheses on $𝖦$ and $k$, wethe tempered dual $\widehat{G}$ of $G$ via the Plancherel formula on $𝔤$, using some character expansions. This involves matching spectral decomposition factors of the Plancherel formulas on $𝔤$ and $G$. As a consequence, we prove that any tempered representation contains a good minimal $𝖪$-type; we extend this result to irreducible...

Let $G$ be a group. If every nontrivial subgroup of $G$ has a proper supplement, then $G$ is called an $aS$-group. We study some properties of $aS$-groups. For instance, it is shown that a nilpotent group $G$ is an $aS$-group if and only if $G$ is a subdirect product of cyclic groups of prime orders. We prove that if $G$ is an $aS$-group which satisfies the descending chain condition on subgroups, then $G$ is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group...

A group $G$ is said to be a $𝒞$-group if for every divisor $d$ of the order of $G$, there exists a subgroup $H$ of $G$ of order $d$ such that $H$ is normal or abnormal in $G$. We give a complete classification of those groups which are not $𝒞$-groups but all of whose proper subgroups are $𝒞$-groups.

For a smooth curve $\Gamma$ and a set $\Lambda$ in the plane ${ℝ}^2$, let $AC(\Gamma ;\Lambda )$ be the space of finite Borel measures in the plane supported on $\Gamma$, absolutely continuous with respect to the arc length and whose Fourier transform vanishes on $\Lambda$. Following [12], we say that $(\Gamma ,\Lambda )$ is a Heisenberg uniqueness pair if $AC(\Gamma ;\Lambda )=\left\{0\right\}$. In the context of a hyperbola $\Gamma$, the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets $\Lambda$ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly...

In this note we study sets of normal generators of finitely presented residually $p$-finite groups. We show that if an infinite, finitely presented, residually $p$-finite group $G$ is normally generated by $g_1,\cdots ,g_k$ with order $n_1,\cdots ,n_k\in \{1,2,\cdots \}\cup \left\{\infty \right\}$, then $${\beta }_1^{\left(2\right)}\left(G\right)\le k-1-\sum _{i=1}^k\frac{1}{n_i},$$ where ${\beta }_1^{\left(2\right)}\left(G\right)$ denotes the first ${\ell }^2$-Betti number of $G$. We also show that any $k$-generated group with ${\beta }_1^{\left(2\right)}\left(G\right)\ge k-1-\epsilon$ must have girth greater than or equal $1/\epsilon$.

A metacyclic group $H$ can be presented as $\langle \alpha ,\beta :{\alpha }^n=1$, ${\beta }^m={\alpha }^t$, $\beta \alpha {\beta }^{-1}={\alpha }^r\rangle$ for some $n$, $m$, $t$, $r$. Each endomorphism $\sigma$ of $H$ is determined by $\sigma \left(\alpha \right)={\alpha }^{x_1}{\beta }^{y_1}$, $\sigma \left(\beta \right)={\alpha }^{x_2}{\beta }^{y_2}$ for some integers $x_1$, $x_2$, $y_1$, $y_2$. We give sufficient and necessary conditions on $x_1$, $x_2$, $y_1$, $y_2$ for $\sigma$ to be an automorphism.

The profinite topology on any abstract group $G$, is one such that the fundamental system of neighborhoods of the identity is given by all its subgroups of finite index. We say that a group $G$ has the Ribes-Zalesskii property of rank $k$, or is RZ${}_k$ with $k$ a natural number, if any product $H_1{H}_2\cdots H_k$ of finitely generated subgroups $H_1,H_2,\cdots ,H_k$ is closed in the profinite topology on $G$. And a group is said to have the Ribes-Zalesskii property or is RZ if it is RZ${}_k$ for any natural number $k$. In this paper we characterize...

It is shown that there exists a $\sigma$-compact topological group which cannot be represented as a continuous image of a Lindelöf $p$-group, see Example 2.8. This result is based on an inequality for the cardinality of continuous images of Lindelöf $p$-groups (Theorem 2.1). A closely related result is Corollary 4.4: if a space $Y$ is a continuous image of a Lindelöf $p$-group, then there exists a covering $\gamma$ of $Y$ by dyadic compacta such that $\left|\gamma \right|\le 2^{\omega }$. We also show that if a homogeneous compact space $Y$ is...

Let $F/k$ be a finite abelian extension of number fields with $k$ imaginary quadratic. Let $O_F$ be the ring of integers of $F$ and $n\ge 2$ a rational integer. We construct a submodule in the higher odd-degree algebraic $K$-groups of $O_F$ using corresponding Gross’s special elements. We show that this submodule is of finite index and prove that this index can be computed using the higher “twisted” class number of $F$, which is the cardinal of the finite algebraic $K$-group $K_{2n-2}\left(O_F\right)$.

See Summary in Note I. First, on the basis of some results in [2] or [5]-such as Lemmas 8.1 and 10.1-the general (mathematical) theorems on controllizability proved in Note I are quickly applied to (mechanic) Lagrangian systems. Second, in case $\mathrm{\Sigma }$, $\chi$ and $M$ satisfy conditions (11.7) when $𝒬$ is a polynomial in $\dot{\gamma }$, conditions (C)-i.e. (11.8) and (11.7) with $𝒬\equiv 0$-are proved to be necessary for treating satisfactorily $\mathrm{\Sigma }$'s hyper-impulsive motions (in which positions can suffer first order discontinuities)....