Variables Separated Polynomials, the Genus 0 Problem and Moduli Spaces

The monodromy method—featuring braid group action—first appeared as a moduli space approach for finding solutions of arithmetic problems that produce reducible variables separated curves. Examples in this paper illustrate its most interesting aspect: investigating the moduli space of exceptions to a specific diophantine outcome. Explicit versions of Hilbert’s irreducibility theorem and Davenport’s problem fostered this technique and motivated the genus 0 problem started by J. Thompson and taken up by many group theorists. We review progress on the genus 0 problem in 0 characteristic, and its quite different contributions in positive characteristic. Example: Let f and h be polynomials with coefficients in a number field K. The classification of finite simple groups shows there is a bound on exceptional degrees for f to the following result. If f is indecomposable and h is not a composition with f , then f(x) − h(y) is irreducible. This answered challenge problems on factorization of variables separated polynomials posed by A. Schinzel in the early 60’s. This limitation result holds, however, only in characteristic 0, one difference between the Genus 0 Problem here and in positive characteristic. Finite field example—Davenport’s Problem: For each finite field Fq there are infinitely many surprising polynomial pairs (f, h) (of degree prime to the characteristic) whose value sets are equal over Fqt , t = 1, 2, . . . . Though we include unpublished results from 30 years ago, a new set of problems stretch the methods. Example of a general theme: Let Md be the elements of Q̄ of degree no more than d over Q. The degree d reducibility set of f , Rf (d) is Rf (d) = {z0 ∈ Md | f(y) − z0 is reducible over Q(z0)}. Similarly, there is a value set Vf (d) (degree 1 fiber). Following a reduction due to G. Frey, for any integer d, there are polynomials f of unbounded degree satisfying Rf (d) \ Vf (d) is finite. The full monodromy method finds precise arithmetic information by extending observations from modular curves. A semi-classical observation: Divisors with support in cusp points on modular curves generate a torsion group on the Jacobian of the curve. Illustrations here (from alternating group covers) show generalizations of this issue are ubiquitous with the monodromy method. Date: October 25, 2001. 1991 Mathematics Subject Classification. Primary 12F05; Secondary 14H40, 14D20, 14E20.