Abstract

Suppose C is an algebraic curve, f is a rational function on C defined over Q, and A is a fractional ideal of Q. If f is not equivalent to a polynomial, then Siegel’s theorem gives a necessary condition for the set C(Q) ∩ f⁻¹(A) to be infinite: C is of genus 0 and the fiber f⁻¹(∞) consists of two conjugate quadratic real points. We consider a converse. Let P be a parameter space for a smooth family Φ : T →P× P¹ of (degree n) genus 0 curves over Q. That is, the fiber T of points of T over × P¹ has genus 0 for in P. Assume a Zariski dense set of in P(Q) have fiber Φ⁻¹( ×∞) over ∞ consisting of two conjugate quadratic real points. The family Φ is then a Siegel family. We ask when the conclusion of Siegel’s theorem — Φ(Q) ∩A is infinite — holds for a Zariski dense subset of in P(Q).

We show how braid action on covers and Hurwitz spaces can tackle this. It refines a unirationality criterion for Hurwitz spaces. A particular family, ₁₀Φ′, of degree 10 rational functions, illustrates this. It arises as the exceptional case for a general result on Hilbert’s Irreducibility Theorem. Fried, 1986 says the only indecomposable polynomials f(y) in Q[y] with f(y) − t reducible in Q[y] for infinitely many t in A∖ f(Q) have degree 5. We show the family ₁₀Φ′ satisfies the converse to Siegel’s theorem. Thus, exceptional polynomials of degree 5 in Fried, 1986 do exist.

We suspect this result generalizes, thus codifying arithmetic accidents occurring in ₁₀P′. To illustrate, we’ve cast this paper as a collection of elementary group theory tools for extracting from a family of covers special cases with specific arithmetic properties. Examples of Siegel and Néron families show the eficiency of the tools, though each case leaves a diophantine mystery.

Authors