虛二次域上的Stark猜想

Abstract

這篇論文參考Stark原本的方法，提供了一個在虛二次域上構造Stark unit的方式。 首先介紹Kronecker極限公式。這個公式告訴我們：在虛二次域上，Artin L-函數在零點的微分值，可以寫成橢圓函數帶值在特殊點上。 接著回顧main theorem of complex multiplication及一些Shimura的成果。這些結果可以幫助我們證明：橢圓函數代值在特殊點，實際上可以生成虛二次域上的abelian擴張。 最後證明CM theta函數的distribution relation。從這個關係式，我們可以證明橢圓函數代值在特殊點，其實是虛二次域上abelian擴張的global unit。

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.