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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?)

  • 1. Paul T. Bateman and Roger A. Horn, Primes represented by irreducible polynomials in one variable, Proc. Sympos. Pure Math., Vol. VIII, 119-132, Amer. Math. Soc., Providence, R.I., 1965, MR 0176966 (31:1234).
  • 2. A. O. Bender and O. Wittenberg, A potential analogue of Schinzel's hypothesis for polynomials with coefficients in $ \mathcal{F}_q[t]$, Int. Math. Res. Not., 36, 2005, 2237-2248. MR 2181456 (2006g:11230)
  • 3. E. R. Berlekamp, An analog to the discriminant over fields of characteristic two, J. Algebra, 38, 1976, 2, 315-317, MR 0404197 (53:8000).
  • 4. Pierre Berthelot and Arthur Ogus, Notes on crystalline cohomology, Princeton University Press, Princeton, 1978, vi+243, MR 0491705 (58:10908).
  • 5. V. Bouniakowsky, Sur les diviseurs numériques invariables des fonctions rationnelles entières, Mémoires sc. math. et phys., 6, 1854, 306-329.
  • 6. K. Conrad, Irreducible values of polynomials: a non-analogy, Number Fields and Function Fields - Two Parallel Worlds, 71-85, Progress in Mathematics, 239, Birkhäuser, Basel, 2005. MR 2176587 (2006i:11033)
  • 7. Brian Conrad and Keith Conrad, The Möbius function and the residue theorem, Journal of Number Theory, 110, 2005, 22-36, issn=0022-314X, MR 2114671 (2006b:11153).
  • 8. Brian Conrad and Keith Conrad, Prime specialization in higher genus I, In preparation.
  • 9. Brian Conrad, Keith Conrad and Robert Gross, Prime specialization in higher genus II, In preparation.
  • 10. W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 2, 2, Springer-Verlag, Berlin, 1997, xiv+470, MR 732620 (85k:14004).
  • 11. Alexander Grothendieck Éléments de géométrie algébrique IV$ _4$. Étude locale des schémas et des morphismes de schémas, Inst. Hautes Études Sci. Publ. Math., 32, 1967, 361 pp., MR 0238860 (39:220).
  • 12. G. H. Hardy and J. E. Littlewood, Some problems of Partition Numerorum III: On the expression of a number as a sum of primes, Acta Math., 44, 1923, 1-70.
  • 13. Matsumura Hideyuki, Commutative ring theory, Cambridge Studies in Advanced Mathematics, 8, 2, Cambridge Univ. Press, Cambridge, 1990, xiii+320, MR 879273 (88h:13001).
  • 14. Bjorn Poonen, Squarefree values of multivariable polynomials, Duke Math. J., 118, 2003, 2, 353-373, MR 1980998 (2004d:11094).
  • 15. Richard G. Swan, Factorization of polynomials over finite fields, Pacific J. Math., 12, 1962, 1099-1106, MR 0144891 (26:2432).

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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

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