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3

Basic facts from number theory

To investigate how singular values of elliptic and modular functions are linked to class fields over quadratic imaginary number fields, we need some facts about orders and ideals in quadratic number fields. Furthermore, we will collect the basic definitions and results from class field theory. Here we will use the classical language because this marries well with singular values of elliptic and modular functions that are essentially dependent on ideals.

3.1 Ideal theory of suborders in a quadratic number field

3.1.1 Fractional ideals, integral ideals, proper ideals, regular ideals

Let K be a quadratic imaginary number field. Let d denote the discriminant and K the maximal order of K. K is a free -module of rank ...

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