Belirsiz ikili kuadratik formların Theta fonksiyonları

The general theory encapsulating the relationship between classical theory of theta functions and representation of integers by positive de nite quadratic forms is well established. Indeed, it is the niteness of the set of representations of a given positive integer by a xed quadratic form forces the convergence of the corresponding theta function. These theta functions also possess certain symmetries with respect to both of their variables. In this thesis with the aim of generalizing the theory of theta functions to the inde nite case we consider the analogous question for the case of inde nite binary quadratic forms. By the classical theory of inde nite binary quadratic forms it is well known that the cardinality of the set of number of representations of any integer is in nite unless empty. It was S. Zwegers who de ned the corresponding theta functions. The convergence and symmetries of these new theta functions corresponding to inde nite binary quadratic forms are investigated. The thesis is organized as follows: the rst part discusses 1-1 correspondences between equivalence classes of rank 2 lattices and positive de nite binary quadratic forms. Second chapter is devoted to recalling basic facts around theta functions (their convergence and symmetries) and then discusses two classical uses of theta functions : in obtaining elliptic functions and in obtaining modular forms. The nal chapter of thesis is devoted to the de nition of theta functions corresponding to inde nite binary quadratic forms and their symmetries.