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...

Let $X$ be a complex manifold, $S$ a generic submanifold of $X^{\mathbb{R}}$, the real underlying manifold to $X$. Let $\mathrm{\Omega }$ be an open subset of $S$ with $\partial \mathrm{\Omega }$ analytic, $Y$ a complexification of $S$. We first recall the notion of $\mathrm{\Omega }$-tuboid of $X$ and of $Y$ and then give a relation between; we then give the corresponding result in terms of microfunctions at the boundary. We relate the regularity at the boundary for ${\overline{\partial }}_b$ to the extendability of $CR$ functions on $\mathrm{\Omega }$ to $\mathrm{\Omega }$-tuboids of $X$. Next, if $X$ has complex dimension 2, we give results...

Let $p$ and $q$ be two different prime integers such that $p\equiv q\equiv 3(mod8)$ with $(p/q)=1$, and $l$ a positive odd square-free integer relatively prime to $p$ and $q$. In this paper we investigate the unit groups of number fields $𝕃=\mathbb{Q}(\sqrt{2},\sqrt{p},\sqrt{q},\sqrt{-l})$.

Let p > 3 be a prime, and let Rₚ be the set of rational numbers whose denominator is not divisible by p. Let Pₙ(x) be the Legendre polynomials. In this paper we mainly show that for m,n,t ∈ Rₚ with m≢ 0 (mod p), $P_{[p/6]}\left(t\right)\equiv -(3/p){\sum }_{x=0}^{p-1}((x³-3x+2t)/p)\left(modp\right)$ and $\left({\sum }_{x=0}^{p-1}((x³+mx+n)/p)\right)²\equiv ((-3m)/p){\sum }_{k=0}^{[p/6]}\left(\genfrac{}{}{0pt}{}{2k}{k}\right)\left(\genfrac{}{}{0pt}{}{3k}{k}\right)\left(\genfrac{}{}{0pt}{}{6k}{3k}\right){((4m³+27n²)/\left(12³·4m³\right))}^k\left(modp\right)$, where (a/p) is the Legendre symbol and [x] is the greatest integer function. As an application we solve some conjectures of Z. W. Sun and the author concerning ${\sum }_{k=0}^{p-1}\left(\genfrac{}{}{0pt}{}{2k}{k}\right)\left(\genfrac{}{}{0pt}{}{3k}{k}\right)\left(\genfrac{}{}{0pt}{}{6k}{3k}\right)/m^k\left(modp²\right)$, where m is an integer not divisible by p.

Let $Q$ denote the field of rational numbers. Let $K$ be a cyclic quartic extension of $Q$. It is known that there are unique integers $A$, $B$, $C$, $D$ such that $$K=Q\left(\sqrt{A(D+B\sqrt{D})}\right),$$ where $$A\text{is}\text{squarefree}\text{and}\text{odd},D=B^2+C^2\text{is}\text{squarefree},B>0,C>0,GCD(A,D)=1.$$ The conductor $f\left(K\right)$ of $K$ is $f\left(K\right)=2^l\left|A\right|D$, where $$l=\left\{\begin{array}{c}3,\text{if}D\equiv 2(mod4)\text{or}D\equiv 1(mod4),B\equiv 1(mod2),\hfill \\ 2,\text{if}D\equiv 1(mod4),B\equiv 0(mod2),A+B\equiv 3(mod4),\hfill \\ 0,\text{if}D\equiv 1(mod4),B\equiv 0(mod2),A+B\equiv 1(mod4).\hfill \end{array}\right.$$ A simple proof of this formula for $f\left(K\right)$ is given, which uses the basic properties of quartic Gauss sums.

Let $E$ be a modular elliptic curve over $\mathbb{Q}$ $L^{\left(n\right)}(s,E)$ denote the $n$-th derivative of its Hasse-Weil $L$-series. We estimate the number of twisted elliptic curves $E_d,d\le D$ such that $L^{\left(n\right)}(1,E_d)\ne 0$.

Let $B$ be a quaternion algebra over a number field $k$. To a pair of Hilbert symbols $\{a,b\}$ and $\{c,d\}$ for $B$ we associate an invariant $\rho ={\rho }_R\left(\left[𝒟\right(a,b\left)\right],\left[𝒟\right(c,d\left)\right]\right)$ in a quotient of the narrow ideal class group of $k$. This invariant arises from the study of finite subgroups of maximal arithmetic kleinian groups. It measures the distance between orders $𝒟(a,b)$ and $𝒟(c,d)$ in $B$ associated to $\{a,b\}$ and $\{c,d\}.$ If $a=c$, we compute ${\rho }_R(\left[𝒟(a,b)\right],\left[𝒟(c,d)\right])$ by means of arithmetic in the field $k\left(\sqrt{a}\right).$ The problem of extending this algorithm to the general case leads to studying a finite...

We consider the Pauli operator $\mathbf{H}\left(\mu \right):={\left({\sum }_{j=1}^m{\sigma }_j\left(-i\frac{\partial }{\partial x_j}-\mu A_j\right)\right)}^2+V$ selfadjoint in $L^2({ℝ}^m;{ℂ}^2)$, $m=2,3$. Here ${\sigma }_j$, $j=1,...,m$, are the Pauli matrices, $A:=(A_1,...,A_m)$ is the magnetic potential, $\mu \>0$ is the coupling constant, and $V$ is the electric potential which decays at infinity. We suppose that the magnetic field generated by $A$ satisfies some regularity conditions; in particular, its norm is lower-bounded by a positive constant, and, in the case $m=3$, its direction is constant. We investigate the asymptotic behaviour as $\mu \to \infty$ of the number of the eigenvalues of $\mathbf{H}\left(\mu \right)$ smaller...