arXiv subjects

Ja Kyung Koo

Publications and source records attributed to Ja Kyung Koo.

At least 19 recordsLinked to original sources

arXiv2026

Adelic framed form class groups and explicit class field theory

Let $D$ be a negative discriminant, and let $K=\mathbb{Q}(\sqrt{D})$. Let $\mathcal{Q}(D)$ denote the set of primitive positive definite binary quadratic forms over $\mathbb{Z}$ of discriminant $D$. We introduce the set of adelic framed forms \begin{equation*} \widehat{\mathcal{Q}}(D)= \left\{(Q,\,γ)\in \mathcal{Q}(D)\times\mathrm{SL}_2(\widehat{\mathbb{Z}})~|~ Q\left(γ\begin{bmatrix}1\\0\end{bmatrix}\right)\in \widehat{\mathbb{Z}}^\times\right\} \end{equation*} and its orbit space $\widehat{C}(D)$ under the natural action of $\mathrm{SL}_2(\mathbb{Z})$. We define an explicit adelic analogue of the Gauss-Dirichlet composition law on $\widehat{C}(D)$ and endow $\widehat{C}(D)$ with the quotient topology induced by the subspace topology on $\widehat{\mathcal{Q}}(D)$ inherited from the product topology on $\mathcal{Q}(D)\times\mathrm{SL}_2(\widehat{\mathbb{Z}})$, where $\mathcal{Q}(D)$ is discrete and $\mathrm{SL}_2(\widehat{\mathbb{Z}})$ has its profinite topology. We then prove that there is an isomorphism of topological groups \begin{equation*} \widehat{C}(D)\simeq\mathrm{Gal}\left(K^\mathrm{ab}(\mathfrak{t}^{1/\infty})/K(\mathfrak{t})\right), \end{equation*} where the Galois group is endowed with the Krull topology, $\mathfrak{t}$ is a positive transcendental real number, and $\mathfrak{t}^{1/\infty}=\{\sqrt[N]{\mathfrak{t}}~|~N\geq1\}$. Moreover, we identify an explicitly defined subgroup of $\widehat{C}(D)$ with $\mathrm{Gal}(K^\mathrm{ab}/K)$ and describe the corresponding Galois action on special values of modular functions. In this way, classical Gauss composition, finite-level form class groups, and Shimura reciprocity are brought together within a single adelic framework. Finally, we show that the abstract group structure of $\widehat{C}(D)$ uniquely determines the imaginary quadratic field $K$.

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arXiv2026

On the integrality of modular functions over $\mathbb{Z}[j]$ and Kronecker-type congruences

Let $N$ be a positive integer and let $f$ be a meromorphic modular function of level $N$ with rational Fourier coefficients. For a prime $p$, define a function $f_p$ on the complex upper half-plane $\mathbb{H}$ by \begin{equation*} f_p(τ)=f\left(\fracτ{p}\right)\quad(τ\in\mathbb{H}). \end{equation*} Let $j$ be the elliptic modular function. We show that if $p\equiv 1$ or $-1\Mod{N}$ and $f$ is integral over $\mathbb{Z}[j]$, then \begin{equation*} \frac{1}{p}(f_p^p-f)(f_p-f^p) \end{equation*} is also integral over $\mathbb{Z}[j]$. This result generalizes the classical Kronecker congruence relation for $j$.

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arXiv2025

A simplified algorithmic realization of Galois actions on special values of modular functions

We propose an explicit and practical algorithm for computing Galois conjugates and irreducible polynomials for special values of modular functions evaluated at CM points associated with imaginary quadratic orders. Our approach builds upon the theory of extended form class groups developed by Jung et al., offering a refinement of earlier methods by Stevenhagen and Cho, respectively.

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arXiv2025

Inverse limits of CM points on certain Shimura varieties

Let $N$ be a positive integer, and let $D\equiv0$ or $1\Mod{4}$ be a negative integer. We define the sets $\mathcal{CM}(D,\,Y_1(N)^\pm)$ and $\mathcal{CM}(D,\,Y(N)^\pm)$ as subsets of the Shimura varieties $Y_1(N)^\pm$ and $Y(N)^\pm$, respectively, consisting of CM points of discriminant $D$ that are primitive modulo $N$. By using the theory of definite form class groups, we show that the inverse limits \begin{equation*} \varprojlim_N\,\mathcal{CM}(D,\,Y_1(N)^\pm)\quad\textrm{and}\quad \varprojlim_N\,\mathcal{CM}(D,\,Y(N)^\pm) \end{equation*} naturally inherit group structures isomorphic to $\mathrm{Gal}(K^\mathrm{ab}/\mathbb{Q})$ and $\mathrm{Gal}(K^\mathrm{ab}(t^{1/\infty})/\mathbb{Q}(t))$, respectively, where $K=\mathbb{Q}(\sqrt{D})$ and $t$ is a transcendental number. These results provide an explicit and geometric interpretation of class field theory in terms of inverse limits of CM points on the associated Shimura varieties.

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arXiv2024

Gauss's form class groups and Shimura's canonical models

Let $N$ be a positive integer and $Γ$ be a subgroup of $\mathrm{SL}_2(\mathbb{Z})$ containing $Γ_1(N)$. Let $K$ be an imaginary quadratic field and $\mathcal{O}$ be an order of discriminant $D_\mathcal{O}$ in $K$. Under some assumptions, we show that $Γ$ induces a form class group of discriminant $D_\mathcal{O}$ (or of order $\mathcal{O}$) and level $N$ if and only if there is a certain canonical model of the modular curve for $Γ$ defined over a suitably small number field. In this way we can find an interesting link between two different subjects, which will be useful in the study of certain quadratic Diophantine equations in terms of primes $p$.

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arXiv2024

Class fields and form class groups for solving certain quadratic Diophantine equations

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 models due to Shimura. As its applications, by using such class fields, for a positive integer $n$ we first find primes of the form $x^2+ny^2$ with additional conditions on $x$ and $y$. Second, by utilizing these form class groups, we derive a congruence relation on special values of a modular function of higher level as an analogue of Kronecker's congruence relation.

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arXiv2023

Arithmetic properties of orders in imaginary quadratic fields

Let $K$ be an imaginary quadratic field. For an order $\mathcal{O}$ in $K$ and a positive integer $N$, let $K_{\mathcal{O},\,N}$ be the ray class field of $\mathcal{O}$ modulo $N\mathcal{O}$. We deal with various subjects related to $K_{\mathcal{O},\,N}$, mainly about Galois representations attached to elliptic curves with complex multiplication, form class groups and $L$-functions for orders.

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arXiv2021

Class fields generated by coordinates of elliptic curves

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 some inequalities on special values of modular functions, we show that a single $x$-coordinate of a certain elliptic curve generates $K_\mathfrak{m}$ over $K$.

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arXiv2021

On some $p$-adic Galois representations and form class groups

Let $K$ be an imaginary quadratic field of discriminant $d_K$ with ring of integers $\mathcal{O}_K$. When $K$ is different from $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$, we consider a certain specific model for the elliptic curve $E_K$ with $j(E_K)=j(\mathcal{O}_K)$ which is defined over $\mathbb{Q}(j(E_K))$. In this paper, for each positive integer $N$ we compare the extension field of $\mathbb{Q}$ generated by the coordinates of $N$-torsion points on $E_K$ with the ray class field $K_{(N)}$ of $K$ modulo $N\mathcal{O}_K$. By using this result we investigate the image of a $p$-adic Galois representation attached to $E_K$ for a prime $p$, in terms of class field theory. Second, we construct the definite form class group of discriminant $d_K$ and level $N$ which is isomorphic to $\mathrm{Gal}(K_{(N)}/\mathbb{Q})$.

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arXiv2020

Form class groups and class fields of CM-fields

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 isomorphic to the ray class group of $K$ modulo $N$. Assuming further that the narrow class number of $F$ is one, we construct a class field of the reflex field of $K$ in terms of the singular values of Hilbert modular functions.

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arXiv2019

On some extension of Gauss' work and applications

Let $K$ be an imaginary quadratic field of discriminant $d_K$, and let $\mathfrak{n}$ be a nontrivial integral ideal of $K$ in which $N$ is the smallest positive integer. Let $\mathcal{Q}_N(d_K)$ be the set of primitive positive definite binary quadratic forms of discriminant $d_K$ whose leading coefficients are relatively prime to $N$. We adopt an equivalence relation $\sim_\mathfrak{n}$ on $\mathcal{Q}_N(d_K)$ so that the set of equivalence classes $\mathcal{Q}_N(d_K)/\sim_\mathfrak{n}$ can be regarded as a group isomorphic to the ray class group of $K$ modulo $\mathfrak{n}$. We further present an explicit isomorphism of $\mathcal{Q}_N(d_K)/\sim_\mathfrak{n}$ onto $\mathrm{Gal}(K_\mathfrak{n}/K)$ in terms of Fricke invariants, where $K_\mathfrak{n}$ is the ray class field of $K$ modulo $\mathfrak{n}$. This would be a certain extension of the classical composition theory of binary quadratic forms, originated and developed by Gauss and Dirichlet.

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arXiv2018

On some extension of Gauss' work and applications (II)

Let $K$ be an imaginary quadratic field of discriminant $d_K$, and let $\mathfrak{n}$ be a nontrivial integral ideal of $K$ in which $N$ is the smallest positive integer. Let $\mathcal{Q}_N(d_K)$ be the set of primitive positive definite binary quadratic forms of discriminant $d_K$ whose leading coefficients are relatively prime to $N$. We adopt an equivalence relation $\sim_\mathfrak{n}$ on $\mathcal{Q}_N(d_K)$ so that the set of equivalence classes $\mathcal{Q}_N(d_K)/\sim_\mathfrak{n}$ can be regarded as a group isomorphic to the ray class group of $K$ modulo $\mathfrak{n}$. We further present an explicit isomorphism of $\mathcal{Q}_N(d_K)/\sim_\mathfrak{n}$ onto $\mathrm{Gal}(K_\mathfrak{n}/K)$ in terms of Fricke invariants, where $K_\mathfrak{n}$ is the ray class field of $K$ modulo $\mathfrak{n}$. This would be certain extension of the classical composition theory of binary quadratic forms, originated and developed by Gauss and Dirichlet.

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arXiv2018

On Siegel invariants of certain CM-fields

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 these invariants and provide some numerical examples to show that this invariant really generates the ray class field of a CM-field.

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arXiv2018

Binary quadratic forms and ray class groups

Let $K$ be an imaginary quadratic field different from $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. For a positive integer $N$, let $K_\mathfrak{n}$ be the ray class field of $K$ modulo $\mathfrak{n}=N\mathcal{O}_K$. By using the congruence subgroup $\pmΓ_1(N)$, we construct an extended form class group whose operation is basically the Dirichlet composition, and explicitly show that this group is isomorphic to the Galois group $\mathrm{Gal}(K_\mathfrak{n}/K)$. We also present algorithms to find all form classes and show how to multiply two form classes. As an application, we describe $\mathrm{Gal}(K_\mathfrak{n}^\mathrm{ab}/K)$ in terms of these extended form class groups for which $K_\mathfrak{n}^\mathrm{ab}$ is the maximal abelian extension of $K$ unramified outside prime ideals dividing $\mathfrak{n}$.

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arXiv2017

On the Schertz conjecture

Schertz conjectured that every finite abelian extension of imaginary quadratic fields can be generated by the norm of the Siegel-Ramachandra invariants. We shall present a conditional proof of his conjecture by means of the characters on class groups and the second Kronecker limit formula.

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arXiv2017

On a problem of Hasse and Ramachandra

Let $K$ be an imaginary quadratic field, and let $\mathfrak{f}$ be a nontrivial integral ideal of $K$. Hasse and Ramachandra asked whether the ray class field of $K$ modulo $\mathfrak{f}$ can be generated by a single value of the Weber function. We completely resolve this question when $\mathfrak{f}=(N)$ for an integer $N>1$.

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arXiv2017

Generators for abelian extensions of number fields

Let $U/L$ be a finite abelian extension of number fields. We first construct a universal primitive generator of $U$ over $L$ whose relative trace to any intermediate field $F$ becomes a generator of $F$ over $L$, too. We also develop a similar argument in terms of norm. As its examples we investigate towers of ray class fields over imaginary quadratic fields. And, we further present a new method of finding a normal element for the extension $U/L$.

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arXiv2017

Generation of ring class fields by eta-quotients

We generate ring class fields of imaginary quadratic fields in terms of the special values of certain eta-quotients, which are related to the relative norms of Siegel-Ramachandra invariants. These give us minimal polynomials with relatively small coefficients from which we are able to solve certain quadratic Diophantine equations concerning non-convenient numbers.

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