Hyperelliptic Modular Curves X * 0 (n ) with Square-free Levels

WN ′ = W (N) N ′ denotes the corresponding Atkin–Lehner involution defined for Γ0(N). (If N ′ = 1, W1 means the identity operator.) Then we define the modular group Γ ∗ 0 (N) to be Γ ∗ 0 (N) = 〈Γ0(N) ∪ {WN ′}N ′ ‖N 〉, i.e., Γ ∗ 0 (N) is generated by Γ0(N) and {WN ′}N ′ ‖N . Then Γ ∗ 0 (N) is a normalizer of Γ0(N) in GL + 2 (Q) = {A ∈M2(Q) | detA > 0}. The factor group Γ ∗ 0 (N)/Γ0(N) is abelian of type (2, . . . , 2) and of order 2 ω(N), where ω(N) denotes the number of distinct prime divisors of N . Moreover, it is known that Γ ∗ 0 (N) is the full normalizer of Γ0(N) if N is divisible neither by 4 nor by 9. In the case N is divisible by 4 or 9, the full normalizer of Γ0(N) is strictly bigger than Γ ∗ 0 (N), and the factor group is no longer abelian. See [1] and [11] for this topic. Let X∗ 0 (N) be the modular curve which corresponds to Γ ∗ 0 (N), namely, X∗ 0 (N) = X0(N)/〈{WN ′}N ′ ‖N 〉.