Related papers: Generation of ring class fields by eta-quotients

We first generate ray class fields over imaginary quadratic fields in terms of Siegel-Ramachandra invariants, which would be an extension of Schertz's result. And, by making use of quotients of Siegel-Ramachandra invariants we also…

We show by adopting Schertz's argument with the Siegel-Ramachandra invariant that singular values of certain quotients of the $\Delta$-function generate ring class fields over imaginary quadratic fields.

We show that the Siegel-Ramachandra invariants could be primitive generators of the ray class fields of imaginary quadratic fields. By using Shimura's reciprocity law we give a modern explanation of the solution of class number one problem…

We first normalize the derivative Weierstrass $\wp'$-function appearing in Weierstrass equations which give rise to analytic parametrizations of elliptic curves by the Dedekind $\eta$-function. And, by making use of this normalization of…

We first construct Siegel invariants of some CM-fields in terms of special values of theta constants, which would be a generalization of Siegel-Ramachandra invariants of imaginary quadratic fields. And, we further describe Galois actions on…

Let $K$ be an imaginary quadratic field and $\mathcal{O}$ be an order in $K$. We construct class fields associated with form class groups which are isomorphic to certain $\mathcal{O}$-ideal class groups in terms of the theory of canonical…

Let $K$ be an imaginary quadratic field different from $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. For a nontrivial integral ideal $\mathfrak{m}$ of $K$, let $K_\mathfrak{m}$ be the ray class field modulo $\mathfrak{m}$. By using…

We produce an infinite family of imaginary quadratic fields whose ideal class groups have $3$-rank at least $2$.

We determine the conditions under which singular values of multiple $\eta$-quotients of square-free level, not necessarily prime to~6, yield class invariants, that is, algebraic numbers in ring class fields of imaginary-quadratic number…

We show that every modular form on $\Gamma_0(2^n)$ ($n\geq2$) can be expressed as a sum of eta-quotients. Furthermore, we construct a primitive generator of the ring class field of the order of conductor $4N$ ($N\geq1$) in an imaginary…

In the making of origami, one starts with a piece of paper, and through a series of folds along seed points one constructs complicated three-dimensional shapes. Mathematically, one can think of the complex numbers as representing the piece…

In this paper, we construct certain infinite families of imaginary quadratic fields whose class number is divisible by a given positive integer.

The invariant theory of finite groups can connect the coding theory to the number theory. In this paper, under this conformity, we obtain the minimal generators of the rings of E-polynomials constructed from the groups related to…

For a number field $K$, we extend the notion of the ring class field of an order in $K$ [C. Lv and Y. Deng, SciChina. Math., 2015] to that of an arbitrary number ring in $K$. We give both ideal-theoretic and idele-theoretic description of…

We develop a criterion for a normal basis, and prove that the singular values of certain Siegel functions form normal bases of ray class fields over imaginary quadratic fields other than $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$.…

Let $K$ be an imaginary biquadratic field and $K_1$, $K_2$ be its imaginary quadratic subfields. For integers $N>0$, $\mu\geq 0$ and an odd prime $p$ with $\gcd(N,p)=1$, let $K_{(Np^\mu)}$ and $(K_i)_{(Np^\mu)}$ for $i=1,2$ be the ray class…

In this article, we focus on orders in arbitrary number fields, consider their Picard groups and finally obtain ring class fields corresponding to them. The Galois group of the ring class field is isomorphic to the Picard group. As an…

We attach a ring of sequences to each number from a certain class of extremal real numbers, and we study the properties of this ring both from an analytic point of view by exhibiting elements with specific behaviors, and also from an…

We construct, for imaginary quadratic number fields with class number 1, an arithmetic site of Connes-Consani type. The main difficulty here is that the constructions of Connes and Consani and part of their results strongly rely on the…

We present two distinct families of imaginary biquadratic fields, each of which contains infinitely many members, with each member having large class groups. Construction of the first family involves elliptic curves and their quadratic…