Prime specialization in genus 0

Abstract:

For a prime polynomial $f(T) \in \mathbf {Z}[T]$, a classical conjecture predicts how often $f$ has prime values. For a finite field $\kappa$ and a prime polynomial $f(T) \in \kappa [u][T]$, the natural analogue of this conjecture (a prediction for how often $f$ takes prime values on $\kappa [u]$) is not generally true when $f(T)$ is a polynomial in $T^p$ ($p$ the characteristic of $\kappa$). The explanation rests on a new global obstruction which can be measured by an appropriate average of the nonzero Möbius values $\mu (f(g))$ 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 $\kappa [u][T]$ does not take prime values as often as analogies with $\mathbf {Z}[T]$ suggest, and it leads to a modified conjecture for how often prime values occur.