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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, Amer. Math. Soc., Providence, R.I., 1965, pp. 119–132. MR 0176966
  • 2. Andreas O. Bender and Olivier Wittenberg, A potential analogue of Schinzel’s hypothesis for polynomials with coefficients in 𝔽_{𝕢}[𝕥], Int. Math. Res. Not. 36 (2005), 2237–2248. MR 2181456, 10.1155/IMRN.2005.2237
  • 3. E. R. Berlekamp, An analog to the discriminant over fields of characteristic two, J. Algebra 38 (1976), no. 2, 315–317. MR 0404197
  • 4. Pierre Berthelot and Arthur Ogus, Notes on crystalline cohomology, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978. MR 0491705
  • 5. V. Bouniakowsky, Sur les diviseurs numériques invariables des fonctions rationnelles entières, Mémoires sc. math. et phys., 6, 1854, 306-329.
  • 6. Keith Conrad, Irreducible values of polynomials: a non-analogy, Number fields and function fields—two parallel worlds, Progr. Math., vol. 239, Birkhäuser Boston, Boston, MA, 2005, pp. 71–85. MR 2176587, 10.1007/0-8176-4447-4_5
  • 7. Brian Conrad and Keith Conrad, The Möbius function and the residue theorem, J. Number Theory 110 (2005), no. 1, 22–36. MR 2114671, 10.1016/j.jnt.2004.06.018
  • 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. William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620
  • 11. A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361 (French). MR 0238860
  • 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. Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
  • 14. Bjorn Poonen, Squarefree values of multivariable polynomials, Duke Math. J. 118 (2003), no. 2, 353–373. MR 1980998, 10.1215/S0012-7094-03-11826-8
  • 15. Richard G. Swan, Factorization of polynomials over finite fields, Pacific J. Math. 12 (1962), 1099–1106. MR 0144891

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Additional Information

Brian Conrad
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043
Email: bdconrad@umich.edu

Keith Conrad
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
Email: kconrad@math.uconn.edu

Robert Gross
Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467-3806
Email: gross@bc.edu

DOI: https://doi.org/10.1090/S0002-9947-08-04283-9
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