Prime specialization in genus 0

Abstract

Abstract. 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.

Prime specialization in higher genus I

“…In characteristic 2, similar periodicity results are described in [2] in terms of more intricate methods resting on 2-adic liftings.…”

“…This paper establishes a higher-genus generalization of theorems proved for the affine line in [2]. To motivate what we will do here, which otherwise may seem idiosyncratic, we begin by reviewing the main conclusions in [2] and two interesting applications.…”

“…The starting point for this paper was the following question: does the periodicity in Theorem 1.1 carry over when κ [u] is replaced with a higher-genus coordinate ring? Since the proofs in [2] for the case of κ[u] made essential use of properties of κ[u] that do not hold in higher genus (such as the absence of Weierstrass gaps at ∞), answering this question is not merely an exercise in adapting the earlier proofs to a more general setting. Our affirmative answer to this motivating question is given in Theorem 1.3, below which we describe some of its applications.…”

“…To motivate what we will do here, which otherwise may seem idiosyncratic, we begin by reviewing the main conclusions in [2] and two interesting applications.…”

“…The main discovery in [2], which is joint work with R. Gross, is that there is a systematic and heuristically reasonable way to predict these new correction factors in advance, in terms of the following feature of the Möbius function on κ [u]. .…”