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.
복소곱 성질을 갖는 타원곡선 E의 성질을 연구함으로써, K가 허이차체일때 타원곡선의 허수승법론을 이용하여 K의 절대유체(Hilbert class field)를 만드는 방법에 대해서 다룬다.
그리고 타원곡선의 양지표(Grossencharacter)를 허수승법론의 주정리를 이용하여 정의하고 이를 이용하여 타원곡선의 L-급수의 성질을 증명한다.
