On the analytic theory of isotropic ternary quadratic forms II

TL;DR

This paper provides an elementary proof for counting automorphic orbits of primitive zeros in certain ternary quadratic forms, refining previous asymptotic results and enhancing understanding of their automorphic properties.

Contribution

It offers a new, elementary proof for a specific case of counting automorphic orbits, complementing prior complex methods and refining asymptotic estimates.

Findings

Elementary proof for special forms

Refinement of asymptotic counts

Corollaries for specific quadratic forms

Abstract

In the first part of this work \cite{Du}, a quantitative supplement to the Hasse principle was given for the count of the number of automorphic orbits of primitive zeros of a genus of ternary quadratic forms. This sequel contains, for certain special forms, an independent and elementary proof of this result. When combined with other results of \cite{Du}, this proof also leads to a refinement of an asymptotic result of \cite{Du} and some corollaries for these special forms.