Prime specialization in genus 0

Abstract

For a prime polynomial f(T) ∈ Z[T], a classical conjecture predicts how often f has prime values. For a finite field k and a prime polynomial f(T) ∈ k[u][T], the natural analogue of this conjecture (a prediction for how often f takes prime values on K[u]) is not generally true when f(T) is a polynomial in T P (p the characteristic of k). The explanation rests on a new global obstruction which can be measured by an appropriate average of the nonzero values μ(f(g)) as g varies. We prove the surprising fact that this Mobius average, which can be defined without reference to any conjectures, has a periodic behavior governed by the geometry of the plane curve f = 0. The periodic average behavior implies in specific examples that a polynomial in κ[u][r] does not take prime values as often as analogies with Z[T] suggest, and it leads to a modified conjecture for how often prime values occur.