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

Ho Yun Jung, Ja Kyung Koo, Dong Hwa Shin, Dong Sung Yoon

Abstract

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.

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

Abstract

Let be an imaginary quadratic field and be an order in . We construct class fields associated with form class groups which are isomorphic to certain -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 we first find primes of the form with additional conditions on and . 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.
Paper Structure (8 sections, 4 theorems, 130 equations, 6 figures)

This paper contains 8 sections, 4 theorems, 130 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Ideal class groups for orders
  3. Generation of class fields
  4. Form class groups of level $N$
  5. Form class groups as Galois groups
  6. Primes of the form $x^2+ny^2$ with some additional conditions
  7. Examples of minimal polynomials over $\mathbb{Q}$
  8. An analogue of Kronecker's congruence relation

Key Result

Theorem A

The map is a well-defined bijection, and hence the group $\Gamma_G$ induces a form class group of level $N$(in the sense of Definition induce).

Figures (6)

  • Figure 1: Homomorphisms of $\mathcal{O}$-ideal class groups
  • Figure 3: A commutative diagram for surjectivity of $\psi_{\Gamma,\,\widetilde{P}_\Gamma}$
  • Figure 4: A commutative diagram for injectivity of $\psi_{\Gamma,\,\widetilde{P}_\Gamma}$
  • Figure 5: A commutative diagram for surjectivity of $\phi$
  • Figure 6: A commutative diagram showing that $\mathcal{C}_\Gamma(D_\mathcal{O},\,N)$ is a form class group
  • ...and 1 more figures

Theorems & Definitions (39)

  • Theorem A: Theorem \ref{['Gformclassgroup']}
  • Theorem B: Corollary \ref{['phiOG']}
  • Theorem C: Lemma \ref{['equivalence']}, Theorem \ref{['x^2+ny^2']}
  • Theorem D: Theorem \ref{['congruence']}
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 29 more