j不变量生成Hilbert类域的简单证明(英文)

A simple proof of j-invariant generates the Hilbert class field

Silverman证明了如下定理:设K是一个虚二次域,E是定义在复数域上的一条带复乘的椭圆曲线,其自同态环为K,则K的Hilbert类域等于K(j),其中j是椭圆曲线E的j不变量.本文给出了该定理的一个简单证明.

Silverman proved the following theorem：Let E be an elliptic curve over C with complex multi-plication OK,where OKis the ring of integers of an imaginary quadratic field K ,then the Hilbert class field of K is generated by the j-invariant of E over K .In this paper we give a simple proof of this funda-mental theorem.