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Transactions of the American Mathematical Society

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Chebotarev density theorem in short intervals for extensions of $ \mathbb{F}_q(T)$


Authors: Lior Bary-Soroker, Ofir Gorodetsky, Taelin Karidi and Will Sawin
Journal: Trans. Amer. Math. Soc. 373 (2020), 597-628
MSC (2010): Primary 11T55; Secondary 11N05, 11S20
DOI: https://doi.org/10.1090/tran/7945
Published electronically: October 1, 2019
MathSciNet review: 4042886
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Abstract: An old open problem in number theory is whether the Chebotarev density theorem holds in short intervals. More precisely, given a Galois extension $ E$ of $ \mathbb{Q}$ with Galois group $ G$, a conjugacy class $ C$ in $ G$, and a $ 1\geq \varepsilon >0$, one wants to compute the asymptotic of the number of primes $ x\leq p\leq x+x^{\varepsilon }$ with Frobenius conjugacy class in $ E$ equal to $ C$. The level of difficulty grows as $ \varepsilon $ becomes smaller. Assuming the Generalized Riemann Hypothesis, one can merely reach the regime $ 1\geq \varepsilon >1/2$. We establish a function field analogue of the Chebotarev theorem in short intervals for any $ \varepsilon >0$. Our result is valid in the limit when the size of the finite field tends to $ \infty $ and when the extension is tamely ramified at infinity. The methods are based on a higher dimensional explicit Chebotarev theorem and applied in a much more general setting of arithmetic functions, which we name $ G$-factorization arithmetic functions.


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Additional Information

Lior Bary-Soroker
Affiliation: Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Email: barylior@post.tau.ac.il

Ofir Gorodetsky
Affiliation: Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Email: ofir.goro@gmail.com

Taelin Karidi
Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
Email: tkaridi@caltech.edu

Will Sawin
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
Email: sawin@math.columbia.edu

DOI: https://doi.org/10.1090/tran/7945
Received by editor(s): November 21, 2018
Received by editor(s) in revised form: July 1, 2019
Published electronically: October 1, 2019
Additional Notes: The first author was partially supported by a grant of the Israel Science Foundation. Part of the work was done while the first author was a member of the Simons CRM Scholar-in-Residence Program
The research of the second author was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 320755.
This research was partially conducted during the period the fourth author served as a Clay Research Fellow and partially conducted during the period he was supported by Dr. Max Rössler, the Walter Haefner Foundation, and the ETH Zurich Foundation.
Article copyright: © Copyright 2019 American Mathematical Society