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4

Factorisation of singular values

Modular functions from FN have the nice property that their values at a quadratic imaginary number α are algebraic and generate an abelian extension of K = (α). Therefore these values are called singular values. Some of these values can also be factorised explicitly, which is crucial for determining the generated fields.

4.1 Singular values

In this section we show that singular values are algebraic, and we will also provide some factorisation properties that will be needed later. First, we consider the modular invariant j. Let be a lattice and α1, α2 a -basis of with . We set

whereby j() is well defined, because ...

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