Examples of Families with Dense Sets of Complex Multiplication Fibers

In this chapter we classify all families C → P _n of covers with a pure (1, n) – VHS. Due to Theorem 4.4.4, all these families have a dense set of CM fibers. We say that a pure (1, n) – VHS is primitive, if the (1, n) eigenspace L _j satisfies that $j \in (\mathbb Z/ (m))*$. Otherwise the pure (1, n) – VHS is derived.

In Section 6.1 we give an integral condition for the branch indices d _k of the family C with the fibers given by $y^m = (x - a_1)^{d_1}..... (x - a_n)^{d_n}.$ This integral condition is stronger than the similar integral condition INT of P. Deligne and G. D. Mostow [18]. Thus we call this strong integral condition SINT. We show that this condition is necessary for the existence of a primitive pure (1, n) – VHS. By using this condition, we compute all examples of families C → P ₁ of covers with a primitive pure (1, 1) – VHS in Section 6.2, which will be listed in Section 6.3. By using the list of examples satisfying INT for n > 1 in [18], we give in Section 6.3 the complete lists of families with a primitive pure (1, n) – VHS. In Section 6.3 we give also the complete list of examples with a derived pure (1, n) - VHS, which will be verified in Section 6.4.