Some studies on modular forms and elliptic curves

Abstract

In this paper, three subjects are discussed in four chapters; they are modular forms Hecke operators and arithmetic of elliptic curves. In chapter 1, it is proved that some object manipulated from Eisenstein series $G_{2}$(z) of weight 2, which has analytically bad properties, belongs to $M_{2}(\Gamma_{0}(p))$ for any prime p. In chapter 2, it is proved that the quotient of Dedekind eta function $(\frac{\eta^{p}(z)}{\eta(pz)})^{2}$ belongs to $M_{p-1}(\Gamma_{0}(p))$ for odd prime p by making use of some object manipulated from level N Eisenstein series of weight 2. In chapter 3, we provide a general thoery of Hecke operators on $\Gamma^0(N)$. In chapter 4, it is proved that 8 is not a congruent number by making use of the arithmetic of elliptic curves.