Some studies on modular curves and artin L-functions

Abstract

In this paper, two subjects are discussed; they are modular curves and Artin L-functions. In chapter 1, we provide a genus formula for a modular curve X$^\circ$(N). Moreover, we investigate how the two curves X$^\circ$(N) and X$_o$(N) are related. In chapter 2, it is proved that the Artin L-function for A$_4$ is holomorphic. For this, the character approach is used. Even though the two subjects mentioned above are dealt with separately analytically, they are closely related with a transform so called Mellin transform.