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
DOI:
https://doi.org/10.1090/S0002-9947-08-04283-9
Published electronically:
January 30, 2008
MathSciNet review:
2379779
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: For a prime polynomial , a classical conjecture predicts how often
has prime values. For a finite field
and a prime polynomial
, the natural analogue of this conjecture (a prediction for how often
takes prime values on
) is not generally true when
is a polynomial in
(
the characteristic of
). The explanation rests on a new global obstruction which can be measured by an appropriate average of the nonzero Möbius values
as
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
.
The periodic Möbius average behavior implies in specific examples that a polynomial in does not take prime values as often as analogies with
suggest, and it leads to a modified conjecture for how often prime values occur.
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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