## Quadratic forms and its applications

Institution Of The Thesis: Bursa Uludağ University, Fen Bilimleri Enstitüsü, Fen Bilimleri Enstitüsü, Turkey

Approval Date: 2009

Thesis Language: Turkish

Student: Arzu Özkoç

Supervisor: Osman Bizim

Abstract:

In this thesis, we consider quadratic forms, and the relationship between elliptic curves, cubic congruances, quadratic ideals, conics and moduler forms.In the first section, we give some definitions, notations and properties which we need in later sections.In the second section, we consider elliptic curves, conics and cubic congruencies over finite fields associated with indefinite binary quadratic forms in the proper cycle of F =(1,7,-6) . We will determine the number of rational points on elliptic curves and conics over F_73 . Moreover, we consider the number of integer solutions of cubic congruences associated with these forms.In the third section, we consider some properties of positive definite binary quadratic forms in a special family. Also we determine the number of integer solutions of quadratic congruencies and determine the number of rational points on singular curves related to forms over finite fields.In the fourth section, we consider the quadratic forms F_1=x_1^2+ 8x_2^2 and G_1=2x_1^2+ 4x_2^2 of discriminant -31 , and their direct sums F_4, G_4, F_3 + G_1, F_2 + G_2 , F_31+ G_3 . We obtain some results concerning the modular forms. Using these, we construct a basis for the cusp form space S_4( (31), 1) , and then we give formulas for the number of representations of positive integer by these quadratic forms and their direct sums.In the last section, for delta = D^1/2 and delta =(1+ D^1/2)/2 values we obtain some results and connection between quadratic irrationals, quadratic ideals and quadratic forms.