TY - JOUR

T1 - NON-VANISHING of L-FUNCTIONS for CYCLOTOMIC CHARACTERS in FUNCTION FIELDS

AU - Lee, Jungyun

AU - Lee, Yoonjin

N1 - Funding Information:
The first author was supported by the National Research Foundation of Korea (NRF) grant founded by the Korean government (NRF-2017R1A6A3A11030486) and 2019 Research Grant from Kangwon National University, and the second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2019R1A6A1A11051177) and also by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST)(NRF-2017R1A2B2004574).
Publisher Copyright:
2021 American Mathematical Society

PY - 2022

Y1 - 2022

N2 - In the number field case, it is conjectured that the central values L(12 , χ) of L-functions are nonzero, where χ : (Z/mZ)∗ → C∗ is a primitive Dirichlet character with conductor m. We resolve this conjecture in the function field case by proving that there are infinitely many cyclotomic characters for which the central values of L-functions are nonzero. In detail, for a given positive integer n, we compute the mean value of L(12 , ηχn) and that of L(12 , χn) for χn ∈ On, where f is a monic irreducible polynomial in A = Fq[t], Fq is the finite field of characteristic p, χn : (A/fnA)∗ → C∗ is a character with some p-power order, On is the set of all the primitive cyclotomic characters χn modulo fn with p-power order, g is a monic polynomial in A that is relatively prime to f, and η : (A/gA)∗ → C∗ is a primitive even character.

AB - In the number field case, it is conjectured that the central values L(12 , χ) of L-functions are nonzero, where χ : (Z/mZ)∗ → C∗ is a primitive Dirichlet character with conductor m. We resolve this conjecture in the function field case by proving that there are infinitely many cyclotomic characters for which the central values of L-functions are nonzero. In detail, for a given positive integer n, we compute the mean value of L(12 , ηχn) and that of L(12 , χn) for χn ∈ On, where f is a monic irreducible polynomial in A = Fq[t], Fq is the finite field of characteristic p, χn : (A/fnA)∗ → C∗ is a character with some p-power order, On is the set of all the primitive cyclotomic characters χn modulo fn with p-power order, g is a monic polynomial in A that is relatively prime to f, and η : (A/gA)∗ → C∗ is a primitive even character.

KW - Central value

KW - Cyclotomic character

KW - Function field

KW - L-function

KW - Mean value

UR - http://www.scopus.com/inward/record.url?scp=85121983643&partnerID=8YFLogxK

U2 - 10.1090/proc/15144

DO - 10.1090/proc/15144

M3 - Article

AN - SCOPUS:85121983643

VL - 150

SP - 455

EP - 468

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 2

ER -