Notes on class field theory and complex multiplication
نویسنده
چکیده
2 Class field theory 2 2.1 Number fields and their completions . . . . . . . . . . . . . . . 2 2.1.1 Number fields, prime ideals . . . . . . . . . . . . . . . . 2 2.1.2 Fractional Ideals . . . . . . . . . . . . . . . . . . . . . . 3 2.1.3 Completions . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.4 Adeles and ideles . . . . . . . . . . . . . . . . . . . . . . 4 2.1.5 Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.6 The trace and the norm . . . . . . . . . . . . . . . . . . 6 2.2 Main theorems of class field theory . . . . . . . . . . . . . . . . 7 2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.1 Cyclotomic fields and the Kronecker-Weber theorem . . 10 2.3.2 Quadratic fields and quadratic reciprocity . . . . . . . . 11
منابع مشابه
Class Field Theory for Number Fields and Complex Multiplication
We state the main results of class field theory for a general number field, and then specialize to the case where K is imaginary quadratic. By looking at elliptic curves with EndC(E) ∼= OK , i.e. E with complex multiplication by OK , we determine the Hilbert class field and ray class fields of K.
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