Prime Specialization in Genus 0

For a prime polynomial f(T ) ∈ Z[T ], a classical conjecture predicts how often f has prime values. For a finite field κ and a prime polynomial f(T ) ∈ κ[u][T ], the natural analogue of this conjecture (a prediction for how often f takes prime values on κ[u]) is not generally true when f(T ) is a polynomial in T p (p the characteristic of κ). The explanation rests on a new global obstruction which can be measured by an appropriate average of the nonzero Möbius values μ(f(g)) as g varies. We prove the surprising fact that this “Möbius 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 Möbius average behavior implies in specific examples that a polynomial in κ[u][T ] 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.