The mean value of the class numbers of cubic function fields

Jungyun Lee; Yoonjin Lee; Jinjoo Yoo

doi:10.1016/j.jmaa.2022.126582

The mean value of the class numbers of cubic function fields

Jungyun Lee, Yoonjin Lee, Jinjoo Yoo

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We compute the mean value of |L(s,χ)|² evaluated at s=1 when χ goes through the primitive cubic Dirichlet odd characters of A:=F_q[T], where F_q is a finite field with q elements and q≡1(mod3). Furthermore, we find the mean value of the class numbers for the cubic function fields K_m=k(m3), where k:=F_q(T) is the rational function field, m∈A is a cube-free polynomial, and deg⁡(m)≡1(mod3).

Original language	English
Article number	126582
Journal	Journal of Mathematical Analysis and Applications
Volume	517
Issue number	1
DOIs	https://doi.org/10.1016/j.jmaa.2022.126582
State	Published - 1 Jan 2023

Keywords

Cubic function field
L-function
Mean value of class number
Moment over function field

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10.1016/j.jmaa.2022.126582

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title = "The mean value of the class numbers of cubic function fields",

abstract = "We compute the mean value of |L(s,χ)|2 evaluated at s=1 when χ goes through the primitive cubic Dirichlet odd characters of A:=Fq[T], where Fq is a finite field with q elements and q≡1(mod3). Furthermore, we find the mean value of the class numbers for the cubic function fields Km=k(m3), where k:=Fq(T) is the rational function field, m∈A is a cube-free polynomial, and deg⁡(m)≡1(mod3).",

keywords = "Cubic function field, L-function, Mean value of class number, Moment over function field",

author = "Jungyun Lee and Yoonjin Lee and Jinjoo Yoo",

note = "Funding Information: J. Lee is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1I1A3057692 ). Y. Lee is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2019R1A6A1A11051177 ) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST)(NRF- 2022R1A2C1003203 ). J. Yoo is a corresponding author, and supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A4A1016649 ) and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2021R1I1A1A01047765 ). Publisher Copyright: {\textcopyright} 2022 Elsevier Inc.",

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doi = "10.1016/j.jmaa.2022.126582",

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AU - Lee, Jungyun

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AU - Yoo, Jinjoo

N1 - Funding Information: J. Lee is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1I1A3057692 ). Y. Lee is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2019R1A6A1A11051177 ) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST)(NRF- 2022R1A2C1003203 ). J. Yoo is a corresponding author, and supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A4A1016649 ) and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2021R1I1A1A01047765 ). Publisher Copyright: © 2022 Elsevier Inc.

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Y1 - 2023/1/1

N2 - We compute the mean value of |L(s,χ)|2 evaluated at s=1 when χ goes through the primitive cubic Dirichlet odd characters of A:=Fq[T], where Fq is a finite field with q elements and q≡1(mod3). Furthermore, we find the mean value of the class numbers for the cubic function fields Km=k(m3), where k:=Fq(T) is the rational function field, m∈A is a cube-free polynomial, and deg⁡(m)≡1(mod3).

AB - We compute the mean value of |L(s,χ)|2 evaluated at s=1 when χ goes through the primitive cubic Dirichlet odd characters of A:=Fq[T], where Fq is a finite field with q elements and q≡1(mod3). Furthermore, we find the mean value of the class numbers for the cubic function fields Km=k(m3), where k:=Fq(T) is the rational function field, m∈A is a cube-free polynomial, and deg⁡(m)≡1(mod3).

KW - Cubic function field

KW - L-function

KW - Mean value of class number

KW - Moment over function field

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JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

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ER -

The mean value of the class numbers of cubic function fields

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Keywords

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