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

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.