On the Integral Representation of Binary Quadratic Forms and the Artin Condition

TL;DR

This paper establishes that the Artin condition, derived from the Artin reciprocity map, is the sole obstruction to the local-global principle for certain integral binary quadratic form equations, supported by concrete examples.

Contribution

It introduces the Artin condition as the unique obstruction to the local-global principle for specific quadratic Diophantine equations, linking class field theory to integral solutions.

Findings

Artin condition is the only obstruction to the local-global principle.

Concrete examples illustrate the application of the Artin condition.

The results connect class field theory with integral solutions of quadratic forms.

Abstract

For diophantine equations of the form ax^2+bxy+cy^2+g=0 over Z whose coefficients satisfy some assumptions, we show that a condition with respect to Artin reciprocity map, which we call the Artin condition, is the only obstruction to the local-global principle for integral solutions of the equation. Some concrete examples are presented.