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

2023/08/22 by Ho Yun Jung, Ja Kyung Koo, Jung, Ho Yun +5 · 2 citations
Engineering · Mathematics · #Soil, Finite Element Methods #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2308.11250

Abstract

Let K be an imaginary quadratic field and O be an order in K. We construct class fields associated with form class groups which are isomorphic to certain 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 x²+ny² 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.

Cited by

• A simplified algorithmic realization of Galois actions on special values of modular functions
• Inverse limits of CM points on certain Shimura varieties

Related

• Gauss's form class groups and Shimura's canonical models
• Form class groups and class fields of CM-fields
• Arithmetic properties of orders in imaginary quadratic fields
• Relations in the 2-Class Group of Quadratic Number Fields
• Generation of ring class fields by eta-quotients