Singular Points on Moduli Spaces and Schinzel’s Problem

Many problems start with two (compact Riemann surface) covers f : X → Pz and g : Y → Pz of the Riemann sphere, Pz, uniformized by a variable z. Some data problems have f and g defined over a number field K, and ask: What geometric relation between f and g hold if they map the values X(OK/p) and Y (OK/p) similarly for (almost) all residue classes of OK/p. Variants on Davenport’s problem interpret as relations between zeta functions [Fr12a, §7.3]. Schinzel’s original problem was to describe expressions f(x) − g(y), with f, g ∈ C(x) nonconstant, that are reducible. The easiest special cases are, as with both Davenport and Schinzel, when the f and g are polynomials (in K[x]). Then, when f is indecomposable, the solutions of both problems are related and they interpret using parameter (Hurwitz) spaces of r-branch point covers. The difficulty: Dropping indecomposability requires dealing with imprimitive groups. That requires group theory beyond the simple group classification. To solve generalizations of the AGZ version of Schinzel’s problem we must go beyond this limitation. Hurwitz spaces characterize the appropriate covers succinctly. Our main formula interprets covers fixed by a Möbius transformation in terms of branch cycles. This describes singular points on reduced Hurwitz spaces when r > 4, and, when r = 4, it interprets when the mysterious moduli group acts trivially.