Binary forms as sums of two squares and Ch\^atelet surfaces

TL;DR

This paper explores how integral binary forms can be expressed as sums of two squares and applies these findings to verify the Manin conjecture for specific Châtelet surfaces over the rationals.

Contribution

It introduces new methods for representing binary forms as sums of two squares and applies these to prove cases of the Manin conjecture for Châtelet surfaces.

Findings

Representation of binary forms as sums of two squares established

Manin conjecture verified for certain Châtelet surfaces

New techniques developed for analyzing rational points on surfaces

Abstract

The representation of integral binary forms as sums of two squares is discussed and applied to establish the Manin conjecture for certain Ch\^atelet surfaces defined over the rationals.