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Mathematische Annalen

, Volume 374, Issue 1–2, pp 447–474 | Cite as

Primes in short intervals on curves over finite fields

  • Efrat Bank
  • Tyler FosterEmail author
Article
  • 58 Downloads

Abstract

We prove an analogue of the Prime Number Theorem for short intervals on a smooth projective geometrically irreducible curve of arbitrary genus over a finite field. A short interval “of size E” in this setting is any additive translate of the space of global sections of a sufficiently positive divisor E by a suitable rational function f. Our main theorem gives an asymptotic count of irreducible elements in short intervals on a curve in the “large q” limit, uniformly in f and E. This result provides a function field analogue of an unresolved short interval conjecture over number fields, and extends a theorem of Bary-Soroker, Rosenzweig, and the first author, which can be understood as an instance of our result for the special case of a divisor E supported at a single rational point on the projective line.

Notes

Acknowledgements

The authors would like to thank Lior Bary-Soroker and Michael Zieve for many conversations during our work on this paper that were crucial to its success. We also extend a warm thank you to Jeff Lagarias for comments on a draft of the paper and for his assistance in formulating prime density theorems in number fields. The exposition benefited greatly from an invitation to speak at the Palmetto Number Theory Series, held at the University of South Carolina and organized by Matthew Boylan, Michael Filaseta, and Frank Thorne, and from an invitation by Jordan Ellenberg to speak in the Number Theory Seminar at the University of Wisconsin– Madison. We are especially thankful to Brian Conrad for looking closely at an earlier version of this paper, for pointing out a gap in our uniformity argument (now filled), for pointing out that we needed to prove what is now Proposition 3.10, and for suggesting that we use [5, §1, Lemma 1.5] to do so. The authors conducted the research that lead to this paper while at the University of Michigan and while the second author was a visiting researcher at L’Institut des Hautes Études Scientifiques, at L’Institut Henri Poincaré, and at the Max Planck Institute for Mathematics. We thank all four institutions for their hospitality. The first author was partially supported by Michael Zieve’s NSF Grant DMS-1162181. Support for the second author came from NSF RTG Grant DMS-0943832 and from Le Laboratoire d’Excellence CARMIN.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of MichiganAnn ArborUSA
  2. 2.Florida State University Mathematics DepartmentTallahasseeUSA

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