On indefinite integral ternary quadratic forms

TL;DR

This paper addresses two longstanding problems related to indefinite integral ternary quadratic forms, developing new tools to handle high ramification in class sum problems involving diophantine invariants.

Contribution

It provides solutions to two open problems from 1990 concerning indefinite ternary quadratic forms, introducing methods to manage high ramification in related sum problems.

Findings

Resolved two open problems from 1990.

Developed new tools for high ramification analysis.

Enhanced understanding of class sums involving quadratic forms.

Abstract

We resolve two problems pertaining to indefinite integral ternary quadratic forms, one highlighted by Margulis and the other initiated by Serre, both from 1990. To do so we develop tools for dealing with high ramification in problems involving sums over classes of such forms weighted by their diophantine invariants.