Twisted Modular Curves

Frey and Kani put a particular modular space structure on the space of a space of modular curves that resembles a Hurwitz space structure. We follow up on this, simplifying the group theory so that properties of this space are clear. For the rst, Prop. 2.4 shows it is simple to characterize the monodromy group of Frey-Kani covers of degree N (and that it is S N). In anticipation of other problems bringing the appearance of Frey-Kani covers, a strengthened result puts them on par with our free characterization of covers whose branch cycles are four dihedral involutions (x1.1; such covers correspond to a point on the modular curve X 0 (N)). Second: Prop. 3.7 shows the corresponding Hurwitz space is connected; it has one absolutely irreducible component over Q. Lemma 2.3 is crucial for the rst, much easier, part. A strengthened version of it concludes the proof of the second part. 1. Statement of the results Assume N is odd, and label a conjugacy class in S N as of type 2 k if it is the conjugacy class of k disjoint 2-cycles. Let C 3n;n?1;1 be the conjugacy classes of type (2 n ; 2 n ; 2 n ; 2 n?1 ; 2) in S N with n = (N ? 1)=2. Recall also the Nielsen class notation g g g 2 C 3n;n?1;1 to mean in some order the entries of g g g = (g 1 ; : : : ; g 5) are in the respective conjugacy classes. 1.1. Basic notation. It is helpful to call an involution (element of order 2) that xes exactly one integer of an odd degree representation a dihedral involution. The phrase product one condition applied to a branch cycle description g g g to mean the product g 1 g 2 g 5 is one. Let N SN (hg g gi) = N g g g be the normalizer in S N of the group hg g gi. Also, H r is the Hurwitz monodromy group regarded as acting on r-tuples. The subgroup SH r consisting of elements that x the order of the conjugacy classes is also of interest.