# Diophantine equations defined by binary quadratic forms over rational function fields

## TL;DR

This paper investigates Diophantine equations defined by binary quadratic forms over rational function fields, establishing that a specific Artin reciprocity condition is the sole obstruction to the local-global principle for integral solutions.

## Contribution

It identifies the exact Artin reciprocity condition that determines the existence of integral solutions over rational function fields.

## Key findings

- The Artin reciprocity map condition is the only obstruction to the local-global principle.
- The study extends classical results to the setting of rational function fields.
- Provides a criterion for solvability of quadratic form equations over function fields.

## Abstract

We study the ``imaginary" binary quadratic form equations ax^2+bxy+cy^2+g=0 over k[t] in rational function fields, showing that a condition with respect to the Artin reciprocity map, is the only obstruction to the local-global principle for integral solutions of the equation.