**Thesis Type:** Postgraduate

**Institution Of The Thesis:** Uludağ Üniversitesi, Turkey

**Approval Date:** 2017

**Thesis Language:** Turkish

**Student:** ŞEYMA KUTLU

**Supervisor: **AHMET TEKCAN

In this thesis, base points of positive definite forms, basis of , indefinite forms, quadratic ideals and integer solutions of some specific Diophantine equations are considered. In the first section, some preliminary notations, definitions and theorems concerning positive definite forms, equivalence of forms, base points, indefinite forms, quadratic ideals and Diophantine equations are given. In the second section, base points of positive definite forms are considered. First we consider the positive definite form whose base points lie on the line for some integer and later we consider the positive definite form whose base points lie on the circle intersection with the upper-half plane. All problems have been considered by using the groups and the extended Hecke group . In the third section, this is the original part of the thesis; base points of positive definite forms and basis of are discussed. Some new theorems and relations are derived. In the last section, this is the original part of the thesis; indefinite quadratic forms, quadratic ideals and the set of all integer solutions of some specific Diophantine equations derived from the indefinite forms are considered.