Representations of squares by positive ternary quadratic forms

Abstract

In this thesis, we study various properties of representations of squares by ternary quadratic forms. A (positive definite integral) ternary quadratic form is called strongly S-regular if it satisfies a regularity property on the number of representations of squares of integers. We explain the relation between the strongly S-regularity and the conjecture given by Cooper and Lam, and we resolve their conjecture completely. We prove that there are only finitely many strongly S-regular ternary forms up to isometry if the minimum of the non zero squares that are represented by the form is fixed. In particular, we show that there are exactly 207 non-classic integral strongly S-regular ternary quadratic forms representing one.