5

The Reciprocity Law

In Chapter 4 we have shown that singular values of functions from *F*_{N} generate algebraic number fields. Our aim now is to determine the action of automorphisms on these values. The main result is the assertion of Theorem 5.1.2, which tells us that the generated fields are abelian extensions of imaginary quadratic number fields. In the following section we will discuss some applications which we will need later.

5.1 The Reciprocity Law of Weber, Hasse, Söhngen, Shimura

The "source" of the Reciprocity Law 5.1.2 is the following theorem:

**Theorem 5.1.1** *Let K be a quadratic imaginary number field and p a prime that splits in K: p* = . *Let* *be a proper ideal of* _{t}, *α*_{1}, *α*_{2} *a basis of* *and* . *We assume p* *t. Let* *be a primitive matrix ...*

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