On Stark Conjecture for Imaginary Quadratic Fields

In this thesis, we provide a construction of Stark units in the case of imaginary quadratic fields following the original approach of Stark. First, we introduce the Kronecker limit formulas, which show that the derivative of Artin L-function for imaginary quadratic field at s=0 can be written in terms of special values of elliptic functions. We then review the main theorem of complex multiplication and results of Shimura, which enable us to prove special values of elliptic functions actually generate abelian extensions of imaginary quadratic fields. Finally, we prove the distribution relation for special values of CM theta functions, with which we show special values of elliptic functions are indeed global units in abelian extensions of imaginary quadratic fields.