Related papers: Gauss's form class groups and Shimura's canonical …

We study differential equations satisfied by modular forms associated to $\Gamma_1\times\Gamma_2$, where $\Gamma_i (i=1,2)$ are genus zero subgroups of $SL_2(\mathbf R)$ commensurable with $SL_2(\mathbf Z)$, e.g., $\Gamma_0(N)$ or…

The differential caluli $(Gamma,d)$ on quantum groups are classified due to the property of the generating element $X$ of its differential $d$. There are, on the one hand differential caluli which contain this element $X$ in the basis of…

Let $L$ be a complete discrete valuation field of prime characteristic $p$ with finite residue field. Denote by $\Gamma _{L}^{(v)}$ the ramification subgroups of $\Gamma _{L}=\operatorname{Gal}(L^{sep}/L)$. We consider the category…

Let $Q(x_1, \cdots,x_n)$ be a real indefinite quadratic form of the type $(r,s)$, $n=r+s$, signature $\sigma=r-s$ and determinant $D\neq 0$. Let $\Gamma_{r,n-r}$ denote the infimum of all numbers $\Gamma$ such that for any real numbers…

Let $(K,\mathcal O,k)$ be a $p$-modular system and assume $k$ is algebraically closed. We show that if $\Lambda$ is an $\mathcal O$-order in a separable $K$-algebra, then $\textrm{Pic}_{\mathcal O}(\Lambda)$ carries the structure of an…

Let $F$ be a totally real number field of class number one, and let $K$ be a CM-field with $F$ as its maximal real subfield. For each positive integer $N$, we construct a class group of certain binary quadratic forms over $F$ which is…

Let $\Gamma$ be a nonelementary discrete subgroup of SU(n,1) or Sp(n,1). We show that if the trace field of $\Gamma$ is contained in $\mathbb R$, $\Gamma$ preserves a totally geodesic submanifold of constant negative sectional curvature.…

We study strongly graded groupoids, which are topological groupoids $\mathcal G$ equipped with a continuous, surjective functor $\kappa: \mathcal G \to \Gamma$, to a discrete group $\Gamma$, such that $\kappa^{-1}(\gamma)\kappa^{-1}(\delta)…

Let $D$ be a square-free integer other than 1. Let $K$ be the quadratic field ${\mathbb Q}(\sqrt D)$. Let $\delta \in \{1,2\}$ with $\delta=2$ if $D\equiv 1 \pmod 4$. To each prime ideal $\mathcal P$ in $K$ that splits in $K/\mathbb Q$ we…

Given a discrete group $\Gamma=<g_1,\ldots,g_M>$ and a number $K\in\mathbb N$, a unitary representation $\rho:\Gamma\to U_K$ is called quasi-flat when the eigenvalues of each $\rho(g_i)\in U_K$ are uniformly distributed among the $K$-th…

A group $G$ admits an \textbf{\em $n$-partite digraphical representation} if there exists a regular $n$-partite digraph $\Gamma$ such that the automorphism group $\mathrm{Aut}(\Gamma)$ of $\Gamma$ satisfies the following properties:…

Let $\Gamma_n(\mathcal{\scriptstyle{O}}_\mathbb{K})$ denote the Hermitian modular group of degree $n$ over an imaginary-quadratic number field $\mathbb{K}$. In this paper we determine its maximal discrete extension in $SU(n,n;\mathbb{C})$,…

In this paper, we prove the existence of an efficient algorithm for the computation of $q$-expansions of modular forms of weight $k$ and level $\Gamma$, where $\Gamma \subseteq SL_{2}({\mathbb{Z}})$ is an arbitrary congruence subgroup. We…

Let $K$ be an imaginary quadratic field and let $\mathcal{O}_K$ be its ring of integers. For an integral ideal $\mathfrak{n}$ of $\mathcal{O}_K$, let $\Gamma_0({\mathfrak{n}})$ be the congruence subgroup of level ${\mathfrak{n}}$ consisting…

We fill the gaps in A. Gica's determination of all the odd positive integers $d$ for which the number of distinct prime divisors of $f_d(x)=d+x^2$ is less than or equal to $2$ for all the positive and odd integers $x\leq\sqrt{d}$. We also…

lambda-good frame is for us a parallel of the class of models of a superstable theory. Our main line is to start with lambda-good^+ frame s, categorical in lambda, n-successful for n large enough and try to have parallel of stability theory…

The Gauss decompositions of the quantum groups, related to classical Lie groups and supergroups are considered by the elementary algebraic and $R$-matrix methods. The commutation relations between new basis generators (which are introduced…

For a Dedekind domain $R$ with field of fractions $K$ a classical $R$-order in a semisimple $K$-algebra $A$ is an $R$-projective $R$-subalgebra $\Lambda$ of $A$ such that $K\Lambda=A$. We study differential graded $K$-algebras which are…

The Steinitz class of a number field extension K/k is an ideal class in the ring of integers O_k of k, which, together with the degree [K:k] of the extension determines the O_k-module structure of O_K. We call rt(k,G) the classes which are…

We improve a result of H. L. Montgomery and J. P. Weinberger by establishing the existence of infinitely many fundamental discriminants $d>0$ for which the class number of the real quadratic field $\mathbb{Q}(\sqrt{d})$ exeeds…