Some studies on complex multiplicaton of elliptic curves

Abstract

In chapter 2, we show that the singular j-invariant has special properties. First we prove that j is an algebraic integer. Second, by using j we construct the Hilbert class field of an imaginary quadratic field K. Lastly, we will give an example about some property of j at the end of chaper 1. In chapter 3, we define the Grossencharacter attached to an elliptic curve E/L by using the main theorem of complex multiplication, and we show that if E has complex multiplication, then the L-series of E/L has an analytic continuation to the entire complex plane and satisfies a functional equation by showing that the L-series of E/L is a product of Hecke L-series with the associated Grossencharacter.