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Transactions of the American Mathematical Society

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Prime specialization in genus 0

Authors: Brian Conrad, Keith Conrad and Robert Gross
Journal: Trans. Amer. Math. Soc. 360 (2008), 2867-2908
MSC (2000): Primary 11N32
Published electronically: January 30, 2008
MathSciNet review: 2379779
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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.

References [Enhancements On Off] (What's this?)

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Additional Information

Brian Conrad
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043

Keith Conrad
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009

Robert Gross
Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467-3806

Keywords: Bateman--Horn conjecture, Hardy--Littlewood conjecture, M\"obius function
Received by editor(s): June 19, 2005
Received by editor(s) in revised form: February 11, 2006
Published electronically: January 30, 2008
Article copyright: © Copyright 2008 American Mathematical Society