On the Integral Representation of $ax^2+by^2$ and the Artin Condition

TL;DR

This paper investigates when the Artin condition fully explains the failure of the local-global principle for integral solutions of quadratic equations over number fields, providing specific examples.

Contribution

It demonstrates that for certain parameters, the Artin condition is the sole obstruction to the local-global principle for quadratic equations over number fields.

Findings

Artin condition is the only obstruction in specific cases

Provides concrete examples illustrating the theory

Clarifies the role of the Artin condition in integral solutions

Abstract

Given a number field $F$ with $\o_F$ its ring of integers. For certain $a,b$ and $\alpha$ in $\o_F$, we show that the Artin condition is the only obstruction to the local-global principle for integral solutions of equation $ax^2+by^2=\alpha$. Some concrete examples are presented at last.