TY - JOUR

T1 - Chebotarev density theorem in short intervals for extensions of Fq(T)

AU - Bary-Soroker, Lior

AU - Gorodetsky, Ofir

AU - Karidi, Taelin

AU - Sawin, Will

N1 - Publisher Copyright:
©2019AmericanMathematicalSociety.

PY - 2020

Y1 - 2020

N2 - 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 Q with Galois group G, a conjugacy class C in G, and a 1 ≥ ϵ > 0, one wants to compute the asymptotic of the number of primes x ≤ p ≤ x+xϵ with Frobenius conjugacy class in E equal to C. The level of difficulty grows as ϵ becomes smaller. Assuming the Generalized Riemann Hypothesis, one can merely reach the regime 1 ≥ ϵ > 1/2. We establish a function field analogue of the Chebotarev theorem in short intervals for any ϵ > 0. Our result is valid in the limit when the size of the finite field tends to ∞ 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.

AB - 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 Q with Galois group G, a conjugacy class C in G, and a 1 ≥ ϵ > 0, one wants to compute the asymptotic of the number of primes x ≤ p ≤ x+xϵ with Frobenius conjugacy class in E equal to C. The level of difficulty grows as ϵ becomes smaller. Assuming the Generalized Riemann Hypothesis, one can merely reach the regime 1 ≥ ϵ > 1/2. We establish a function field analogue of the Chebotarev theorem in short intervals for any ϵ > 0. Our result is valid in the limit when the size of the finite field tends to ∞ 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.

UR - http://www.scopus.com/inward/record.url?scp=85077443232&partnerID=8YFLogxK

U2 - 10.1090/tran/7945

DO - 10.1090/tran/7945

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AN - SCOPUS:85077443232

SN - 0002-9947

VL - 373

SP - 597

EP - 628

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 1

ER -