On Orders in Number Fields: Picard Groups, Ring Class Fields and Applications

TL;DR

This paper explores orders in number fields, their Picard groups, and associated ring class fields, providing criteria for solving certain quadratic Diophantine equations over imaginary quadratic fields.

Contribution

It establishes the isomorphism between Galois groups of ring class fields and Picard groups for arbitrary number fields, and applies this to Diophantine equations.

Findings

Galois group of ring class field is isomorphic to the Picard group.

Criteria for solvability of $p=x^2+ny^2$ over specific imaginary quadratic fields.

Framework connecting orders, Picard groups, and class field theory.

Abstract

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 application, we give criteria of the solvability of the diophantine equation $p=x^2+ny^2$ over a class of imaginary quadratic fields where $p$ is a prime element.