Modular forms, elliptic curves and their applications


Thesis Type: Doctorate

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

Approval Date: 2011

Thesis Language: Turkish

Student: İlker İnam

Supervisor: Osman Bizim

Abstract:

In this work, two recent popular theories of mathematics, namely Elliptic Curves and Modular Forms are discussed. The Taniyama-Shimura-Weil Conjecture which is also named as "Modularity Theorem" after proven, connects these two theories and it has various applications. Using the relationship of these theories an open problem on the Theory of Elliptic Curves is solved by Modular Forms Theory. In the first chapter of the work, some of the concepts which will be used later are introduced. In the second chapter, the theory of elliptic curves is introduced and some results for the families of elliptic curves over finite fields are obtained. In the last section of this chapter, the properties of the elliptic curves defined over rational numbers were observed and some applications are given. The third chapter is reserved for modular forms. Integer and half-integer weight forms are introduced and also the definition of Hecke operator is given. This chapter ended with the statement of Modularity Theorem which is mentioned above. The fourth and final chapter of the work which is the essential part of this thesis, the problem of computing the order of the Selmer group of a randomly selected elliptic curve is considered. In this case, there is no solution of this problem in the literature but restricting twist families of elliptic curves, this problem is partly solved by using the fact that modular forms are analytic functions. While this problem is being solved, it is assumed that one of the award-winning conjectures of the mathematics called the Birch and Swinnerton-Dyer Conjecture is true and an important theorem of J. L. Waldspurger is used. The distribution of the computed orders of Selmer groups expressed with a simple function.